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1
A fast-food restaurant gave a “Customer Satisfaction Survey” in which 1500 customers rated how satisfied they were with service. The results are shown below as a table.
Rating Frequency
Extremely Satisfied 234
Satisfied 443
Neutral 246
Dissatisfied 203
Extremely Dissatisfied 374
The relative frequency of people who were dissatisfied with the service is __________.
13.5%
29.5%
20.3%
38.5%
2
The formula for the standard deviation of a sample is:
Select the true statement for the following data set that has a mean of 8:
4, 6, 6, 6, 9, 9, 12, 12
Answer choices are rounded to the hundredths place.
The variance is 2.98 and the standard deviation is 8.86.
The variance is 8.86 and the standard deviation is 7.50.
The variance is 8.86 and the standard deviation is 2.98.
The variance is 7.50 and the standard deviation is 2.98.
3
Let x stand for the number of minutes spent waiting in line for a rollercoaster at an amusement park. 81 people are sampled at a time. The sample mean is 18 minutes and the sample standard deviation is 0.5 minutes.
What is the standard deviation of the population?
18
4.5
0.5
2
4
Mike's electronics store sold the following number of cellphones on each of the seven days of a week.
Day Cell Phones Sold
Sunday 12
Monday 10
Tuesday 9
Wednesday 5
Thursday 14
Friday 10
Saturday 10
The mean number of cell phones sold by the store for the week was __________.
5 phones
70 phones
9 phones
10 phones
5
The average daily rainfall for the past week in the town of Hope Cove is normally distributed, with a mean rainfall of 2.1 inches and a standard deviation of 0.2 inches.
If the distribution is normal, what percent of data lies between 1.9 inches and 2.3 inches of rainfall?
34%
99.7%
68%
95%
6
Choose the statement that correctly describes a normal distribution.
The approximate percent of values lying within two standard deviations of the mean is 47.5%.
Approximately 68% of the values are greater than the mean value.
The approximate percent of values lying within three standard deviations of the mean is 49.85%.
Approximately 68% of the values lie within one standard deviation of the mean.
7
Hannah noted the height of each student in her class and found that the mean height of the students is 56 inches, with a standard deviation of 1.2 inches. The height of one of the students, James, is 59 inches.
What is the z-score for James' height?
-3.6
2.5
3.6
-2.5
8
Jerry graded seven standardized tests with the following scores:
60, 74, 41, 87, 94, 79, 57
Which standardized test score represents the 50th percentile?
57
41
79
74
9
The difference between the standard deviation and the variance of a standard normal distribution is __________.
0
0.5
1
2
10
Which of the following statements about a positively skewed distribution is true?
The distribution of the data tails to the right of the median.
The mean, median, and mode have the same values.
The distribution of the data features two modes.
The distribution of the data tails to the left of the median.
11
Katherine, Jonathan, and Ryan went bowling. Afterwards, two of them decided to make bar graphs to plot their scores.
Who made Graph 2, and why?
Katherine, because she wanted to accurately show each person's score.
Katherine, because she wanted to make the scores appear reasonably close.
Jonathan, because he wanted to make the scores appear very different.
Jonathan, because he wanted to make the scores appear reasonably close.
12
Which of the following is NOT a step used in calculating standard deviation?
Subtracting the value of each data set from the mean.
Calculating the mean of the data set.
Squaring the difference of x - u.
Dividing the sum of each value by the total number of values plus 1.
13
One week, Rachel earned $250. She spent $120 on food, $30 on miscellaneous items, and saved the rest.
If Rachel makes a pie chart showing how she spends her money, the central angle for the food sector would be __________.
187°
144°
90°
173°
14
The first quartile (Q1) value from the above box plot is __________.
29
40
52
33
15
Consider the histogram showing the heights of individuals on a basketball team.
How many players are taller than 78 inches?
12
3
6
4
16
At Jeremy's school, the final grade for his Human Biology course is weighted as follows:
Tests: 50%
Quizzes: 35%
Homework: 15%
Jeremy has an average of 94% on his tests, 78% on his quizzes, and 62% on his homework.
What is Jeremy's weighted average?
83.6%
74.8%
75.6%
78%
17
Ralph records the time it takes for each of his classmates to run around the track one time. As he analyzes the data on the graph, he locates the mean and median time.
Which component of data analysis is Ralph utilizing?
The overall shape of the data
An outlier in the data set
The overall spread of the data
The center of the data set
18
In which of these cases should the median be used?
When data has no outliers
When the data has small variance
When the data has extreme values
When the data has nominal values
19
Which of the following statements is true?
The Central Limit Theorem is applicable only for data sets comprising exactly thirty samples.
The Central Limit Theorem is applicable only for data sets comprising more than thirty samples.
The Central Limit Theorem is applicable only for data sets comprising less than thirty samples.
The Central Limit Theorem is applicable only for data sets comprising thirty or more samples.
20
This chart shows the number of students of different age groups who participated in a quiz.
Which of the following statements about the stack plot is true?
There were more male students in the age group 19-21 than in the age group 15-17.
There were more male students than female students in the age group 19-21.
There were more students in the age group 15-17 than in any other age group.
There were more female students in the age group 17-19 than in any other age group.
21
Select the statement that is TRUE.
The interquartile range is calculated by adding the first quartile with the third quartile.
The interquartile range covers 100% of a data set.
The interquartile range is calculated by subtracting the first quartile from the third quartile.
The interquartile range is the average value of a data set.
22
An outlier is which of the following?
A value that is twice as small as the mean.
A value that is twice as large as the mean.
A value that is outside of the data set.
A value that is significantly higher or lower than most of the values in the data set.
23
The graph in the figure shows the change in the Consumer Price Index over an 11-year period.
Which segment of the graph indicates no change in the Consumer Price Index?
D
A
H
E
24
Consider the times (in seconds) that it took children and adults to solve a Rubik’s cube at a competition.
What does the circled section represent?
One child solved the Rubik's cube in 21.7 seconds.
21 children completed the Rubik's cube in 7 minutes.
One child took 72 seconds to solve the Rubik's cube, while another took 71 seconds to solve it.
21 children completed the Rubik's cube in 7 seconds.
25
The dotplot below shows the height (in cm) of students in a class.
How many students are taller than 140 centimeters?
10
12
8
11
26
Sara wonders what percentage of her students answered at least half of the quiz questions incorrectly.
The relative cumulative frequency of students who earned a score of 21 or higher on the quiz is __________.
18%
68%
32%
16%
MAT 300 Unit 2 Milestone 2 Exam Answer Sophia Course
1
Sara wonders what percentage of her students answered at least half of the quiz questions incorrectly.
The relative cumulative frequency of students who earned a score of 21 or higher on the quiz is __________ %.
16
32
18
68
To get relative frequency of 21 or greater, we need to find the cumulative number of 21 or more. We simply add up any bin that has the number 21 or more, such as the bin that shows scores of 21-25, 26-30, 31-35, and 36-40. This would be 7 + 3 + 4 + 2 = 16. To get relative frequency, we will take this cumulative number and divide by the total number of students.
number of 21 or more / total number of students = 16 / 50 = 0.32 or 32%.
Cumulative Frequency
2
The midterm exam scores obtained by boys and girls in a class are listed in the table below.
What does the circled section represent?
Two boys scored between 80 and 89 marks on the exam.
Twelve boys scored 8 marks on the exam.
Eight boys scored over 10 marks on the exam.
Eight boys scored 12 marks on the exam.
If we recall that the stem and leaf can give us the actual values in the data set, then the circle corresponds to 81 and 82. We can then note that there are two scores from boys between 80 and 89.
Stem-and-Leaf Plot
3
Dave drives to work. While driving the car over nine days, he observes his daily average speed as 45, 62, 44, 70, 59, 66, 54, 63, and 67 miles per hour.
The median speed at which Dave drove to work was __________.
63 miles per hour
59 miles per hour
62 miles per hour
58.89 miles per hour
To get the median we first order the data and take the middle value. The ordered values are: 44, 45, 54, 59, 62, 63, 66, 67, 70. Since there are an odd number (n=9) of values we simply take the middle, which is the 5th observation, which is
62 mph.
Mean, Median, and Mode
4
Which of the following statements is true for an outlier?
It is a data point that is below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR.
It is a data point that is between Q1 + 1.5 x IQR and Q3 - 1.5 x IQR.
It is a data point that is below or above Q1 ± 1.5 x IQR.
It is a data point that is below or above Q3 ± 1.5 x IQR.
To find an outlier we note the lower bound and upper bound for outliers are Q1 - 1.5IQR and Q3 + 1.5IQR respectively.
Outliers and Modified Boxplots
5
Let x stand for the percentage of an individual student's math test score. 64 students were sampled at a time. The population mean is 78 percent and the population standard deviation is 14 percent.
What is the standard deviation of the sampling distribution of sample means?
64
0.22
1.75
14
The standard deviation of the sampling distribution is
Center and Variation of a Sampling Distribution
6
Given the formula above, select the true statement for this data set: 4, 6, 6, 6, 9, 9, 12, and 12, with a mean of 8. Answer choices are rounded to the hundredths place.
The variance is 8.86 and the standard deviation is 7.50.
The variance is 8.86 and the standard deviation is 2.98.
The variance is 2.98 and the standard deviation is 8.86.
The variance is 7.50 and the standard deviation is 2.98.
If we first note the variance of the data is:
If we note that the standard deviation(SD) is simply the square root of the variance, then the SD =
Standard Deviation
7
At Jeremy's school, the final grade for his Human Biology course is weighted as follows:
Tests: 50%
Quizzes: 35%
Homework: 15%
Jeremy has an average of 94% on his tests, 78% on his quizzes, and 62% on his homework.
What is Jeremy's weighted average?
83.6%
78%
75.6%
74.8%
In order to get the weighted average we use the following formula:
Weighted Mean
8
Nick just received his 20 question test back. He scored a 24 out of a possible 60 points. His teacher told him the mean score on the test was a 35, with a standard deviation of 6.
What is Nick's z-score?
4.17
-4.17
1.83
-1.83
Recall the z = (value - mean)/SD = (24-35)/6 = -1.83. So 24 is -1.83 z-scores or SD's below the mean of 35.
Standard Scores and Z-Scores
9
A fast-food restaurant gave a “Customer Satisfaction Survey” in which 1500 customers rated how satisfied they were with service. The results are shown below as a table and pie chart.
Rating Frequency
Extremely Satisfied 234
Satisfied 443
Neutral 246
Dissatisfied 203
Extremely Dissatisfied 374
The relative frequency of people who were dissatisfied with the service is __________%.
29.5
20.3
13.5
38.5
To get the relative frequency, we take the ( number / total ). So in this case for dissatisfied = (number dissatisfied / total) =
(203/1500) = 0.135 or 13.5%.
Frequency Tables
10
When a survey was conducted among 100 students to find their favorite pizza topping, 45 students voted for pepperoni, 25 for mushrooms, and 30 voted for cheese.
If a pie chart were made showing the number of votes for each topping, the central angle for the cheese sector would be __________.
90°
198°
162°
108°
Recall that to get the angle for something in a pie chart we use the formula (value / total value)*360. So in this case, for cheese (value cheese / total ) * 360 = (30/100)*360 = 108 degrees.
Bar Graphs and Pie Charts
11
The graph below shows the change in passenger load factor for all scheduled airlines in the United States over different months of a year.
Which segment of the graph indicates no change in the passenger load?
A
C
B
D
In the graph, when the segment is horizontal, this shows no change in the percentage of passengers. This would indicate no change in passenger load. This is segment B.
Line Charts and Time-Series Diagrams
12
The school paper published the results of a survey among grade eight students with regards to their favorite subject.
Which of the following statements about the stack plot is true?
More girls than boys chose English as their favorite subject.
Most of the students chose English as their favorite subject.
More boys chose math as their favorite subject than science.
More than half the girls chose science as their favorite subject.
When examining English, we can see that the darker area (girls) is much larger than the boys. So it is true that more girls choose English as their favorite subject. In fact, in all subjects it appears that there are more girls relative to boys.
Stack Plots
13
The first quartile (Q1) value from the above box plot is __________.
33
40
29
52
Note the value for Q1 is the left edge of the box, which is 33.
Five Number Summary and Boxplots
14
In which of these cases should mode be used?
When the data has extreme values
When the data is represented using ratio scale
When the data is represented using interval scale
When the data is qualitative and we talk about the most frequent category
If the data is qualitative, it is only descriptive. In this case, the mode is a good measure since the mode examines the most frequently occurring value. The data can be non-numeric.
Measures of Center
15
Which of the following statements is true?
For the Central Limit Theorem to be true, you must have a large sample, the underlying population must be normally distributed, and the standard deviation should not be finite.
For a large enough sample size, the Central Limit Theorem states that the sample means of repeated samples of a population are normally distributed.
For a large enough sample size, the Central Limit Theorem states that the sample medians of repeated samples of a population are normally distributed.
Even with a very large sample size, the Central Limit Theorem states that the sample means of repeated samples of a population cannot be normally distributed.
The Central Limit Theorem gives us information about the properties' sampling distributions of statistics to have given that the sample size is large enough. It tells us the sampling distribution's mean should be equal to the true population mean.
Shape of a Sampling Distribution
16
The dotplot below shows the number of songs Carly downloaded over 15 successive weekends.
What is the greatest number of songs downloaded?
5
8
10
9
The dotplot is a number line that shows the number of items at each value, which is designated with an X. So the largest value we can see is at 10.
Dot Plots
17
For a class reading competition, the students were asked to read a book. Mike, Jack, and Rayon discussed the numbers of pages they read on the first day. One of them made a graph to represent the data.
Who made the graph, and why?
Mike, because he wanted to make the amount read by each person appear reasonably close.
Mike, because he wanted to accurately show the amount read by each person.
Rayon, because he wanted to make the amount read by each person appear very different.
Jack, because he wanted to make it look like he read significantly more than the others.
Since there was a competition, the person who most likely made this graph would want to represent themselves favorably. Since Jack has the most pages, it would probably be him.
Misleading Graphics
18
The mean of any standard normal distribution is __________.
0.5
1
0
±0.5
The standard normal is always centered at 0. It also has a standard deviation of 1.
Standard Normal Distribution
19
Select the false statement about standard deviation.
It is the square root of variance.
It is the average of the squared differences of the values from the mean.
It is calculated using the mean.
It is a measure of how spread out the values of a data set are.
Recall that the standard deviation(SD) = and the variance is the average of the squared distances from the mean. So, the SD is the average distance to the mean.
Standard Deviation
20
The quality control inspector of a factory manufacturing screws found that the samples of screws are normally distributed with a mean length of 5.5 cm and a standard deviation of 0.1 cm.
If the distribution is normal, what percent of data lies between 5.3 centimeters and 5.7 centimeters?
99.7%
34%
95%
68%
Recall if the data is normal it follows the empirical rule. The length of the interval is 0.4. So from the mean of 5.5, this is 0.2 in either direction or two standard deviation. This means that 95% of the data should lie between 5.3 to 5.7.
68-95-99.7 Rule
21
Ralph records the time it takes for each of his classmates to run around the track one time. As he analyzes the data on the graph, he notices very little variation between his classmates’ times.
Which component of data analysis is Ralph observing?
The overall shape of the data
The overall spread of the data
The center of the data set
An outlier in the data set
Since Ralph is looking at the variation of data, this is examining the spread of the data.
Data Analysis
22
Select the statement that is FALSE.
The range is never greater than the greatest value of a data set.
The interquartile range is the difference between the highest and lowest values in the middle of a data set.
The mean is never greater than the greatest value of a data set.
The range is the difference between the largest and smallest values of a data set.
If we recall that the range is the max-min value, in a dataset where the max = 20 and the min = -5, then the range = 20 - (-5) = 25. So, the range can certainly be larger than the max value.
Range and Interquartile Range (IQR)
23
Jerry graded seven standardized tests with scores of 60, 74, 41, 87, 94, 79, and 57.
Which standardized test score represents the 50th percentile?
57
41
74
79
If we note that there are 7 values, so the 50th percentile is the median, which is the 4th value. Make sure to first order the data (41, 57, 60, 74, 79, 87, 94), so the 4th value is 74.
Percentiles
24
Consider the histogram showing the weights of babies born in a hospital over a month.
What is the difference in the weights of the lightest baby and the heaviest baby born in the hospital?
7.5 pounds
5.5 pounds
3 pounds
6 pounds
If we look at the histogram, the heaviest baby is 10 lbs and the lightest baby is about 4.5 lbs. So the difference between the heaviest and lightest baby would be 10 - 4.5 = 5.5 lbs.
Histograms
25
Which of the following statements about a normal distribution is true?
The normal distribution is single-peaked and symmetric.
A large portion of the data is skewed to the right.
A large portion of the data is located near the tails.
The normal distribution is an example of a bimodal distribution.
A normal distribution is a bell-shaped and symmetric distribution. So it has a single smooth peak, which tells us the mean and median are the same.
Normal Distribution
26
Which of the following statements about a positively skewed distribution is true?
The mean, median, and mode have the same values.
The distribution of the data tails to the right of the median.
The distribution of the data features two modes.
The distribution of the data tails to the left of the median.
Skewness refers to how the data trends to the left or right. If a dataset is skewed, it is not symmetric. The direction of the tail of a distribution tells you which direction the skew lies. If there is positive skew, this implies the skew is to the right. If the distribution trends to the right, it will have a mean that is larger than the median due to those higher values.
Shapes of Distributions
© 2019 SOPHIA Learning, LLC. SOPHIA is a registered
1
Erica is performing an experiment that requires her to weigh multiple samples. The masses of her samples are found to be normally distributed with a mean of 157g and a standard deviation of 5.2g.
If Erica wants to convert her data to a standard normal distribution, which of the following statements is true?
The new mean would be 0, and the standard deviation would be 5.
The new mean would be 1, and the standard deviation would be 0.
The new mean would be 0, and the standard deviation would be 1.
The new mean would be 1, and the standard deviation would be 5.
The mean and standard deviation for a standard normal is always 0 and 1. It is standardized since it takes all the raw data and converts them into z-scores.
Standard Normal Distribution
2
The formula for standard deviation is
Find the standard deviation of the following data set that has a mean of 5:
2, 4, 4, 4, 5, 5, 5, 7, 9
1
0
2
4
First, determine the variance of the data:
If we recall the standard deviation (SD) is the square root of the variance, so we can note SD
Standard Deviation
3
Stan is looking at the statistics for his favorite baseball player, who has hit 25, 26, 32, 38, 43, 40, 28, 32, 34, and 42 home runs in ten seasons.
Using this data set, match each value with the correct description.
Mean
Median
Mode
A.
32
B.
33
C.
34
The mode is the most frequently occurring data point. The number 32 occurs two times, which is more than any other number's frequency.
The mean is equal to the sum of the values divided by how many numbers there are. When we add the total number of home runs, we get 340. There were 10 seasons, so the mean can be calculated by the following calculation:
The median is the middle value once the data is ordered.
Since there are an even number of values, we take the average of the 2 middle values of 32 and 34 to get a mean of 33:
Mean, Median, and Mode
4
A workplace gave an “Employee Culture Survey” in which 500 employees rated their agreement with the statement, “I feel respected by those I work for.”
Rating Frequency
Strongly Agree 156
Agree 114
Neutral 99
Disagree 88
Strongly Disagree 43
The relative frequency of people who strongly agree with the statement is __________.
8.6%
31.2%
16%
54%
To get the relative frequency, we take the frequency of the value and divide it by the total number. So in this case for strongly agree, the relative frequency would be:
Frequency Tables
5
Let x stand for the percentage of an individual student's math test score. 100 students were sampled at a time. The population mean is 75 percent and the population standard deviation is 12 percent.
What are the mean and standard deviation of the sampling distribution of sample means?
mean = 75, standard deviation = 12
mean = 7.5, standard deviation = 1.2
mean = 7.5, standard deviation = 12
mean = 75, standard deviation = 1.2
The mean of the sampling distribution should be the true population mean, which would be 75 percent.
The standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size:
Center and Variation of a Sampling Distribution
6
Match each term with its corresponding definition.
A number that describes the middle of a set of data
A qualitative statement about how the data looks after it has been plotted
A number that describes how far the data is from the middle
A.
Shape
B.
Center
C.
Spread
Recall that center describes the middle. Spread tells us how the data is distributed and is generally a measure from center (such as variance and standard deviation). Shape describes what the plotted data looks like and is a qualitative measure because it simply describes the shape.
Data Analysis
7
Consider the histogram showing the heights of individuals on a basketball team.
Which of the following is the difference in height between the shortest player on the team and the tallest player on the team?
4 inches
81 inches
9 inches
75 inches
The shortest height is 75 inches. In the histogram, we can see the frequency is 3, indicating that there are 3 players at that height.
Similarly the tallest height is 84 inches. There are 2 players that tall.
So the difference in height from the largest to the smallest player is from 75 to 84 inches, or 9 inches.
Histograms
8
Which of the following statements is NOT true?
The Central Limit Theorem is applicable only for data sets comprising 30 or more samples.
For the Central Limit Theorem to be true, you must have a large sample, the underlying population must be normally distributed, and the standard deviation should not be finite.
According to the Central Limit Theorem, the mean of the sampling distribution is equal to the population mean.
For a large enough sample size, the Central Limit Theorem states that the sample means of repeated samples of a population are normally distributed.
The Central Limit Theorem (CLT) puts no restrictions on the type of population you draw from. It could be normal, uniform, skewed, etc. So, the CLT does not say you must draw from a normal population. It also requires that the variance is finite, which simply means it must be defined.
Shape of a Sampling Distribution
9
Sara wonders what percentage of her students answered over 75% of the quiz questions correctly.
The relative cumulative frequency of students who earned a score of 31 or higher on the quiz is __________%.
88%
6%
44%
12%
To get the relative frequency of 31 or higher, we need to find the cumulative number of 31 or more. We simply add up any bin that has the number 31 or more, such as the bin that shows scores of 31-35 and 36-40.
This would be:
To get relative frequency, we will take this cumulative number and divide it by the total number of students.
Cumulative Frequency
10
A baseball scout recorded the type of pitch a pitcher threw during a game and whether it was thrown for a strike or a ball.
Which of the following is a true statement about the stacked bar chart?
The pitcher threw under 100 total pitches.
The pitcher threw about the same number of strikes when throwing changeups as he did throwing fastballs.
The pitcher’s most accurate pitch (highest percent strikes) is the curveball.
Over half of the sliders the pitcher threw were balls.
Recall that strikes are the lighter color. So if we look at the lighter part of each graph, we can note strikes for changeup is 10 to 35 or about 25 strikes. For fastball it is 22 to 47 or 25 strikes. So there are about the same number of strikes for both of these pitches.
Stack Plots
11
The graph in the figure shows the Gross Domestic Product (GDP) from 2008-2011.
The segment of the graph that corresponds to almost no GDP growth at all is __________.
C
A
D
B
In this graph, a segment that is horizontal would show no change in the consumer price index. This occurs at segment C.
Line Charts and Time-Series Diagrams
12
Joe is playing a game in which he has to roll two six-sided dice. In his past ten rolls, he has rolled a sum of one 2, two 5s, three 7s, two 8s, one 10, and one 12.
The weighted mean of all of Joe’s dice rolls is __________.
6.8
7.3
7.5
7.1
In order to get the weighted average, we use the following formula:
The weight for each value is the number of times a value counts towards the total. For example, the value 2 occurred once, the value of 5 occurred twice, etc.
Weighted Mean
13
Which of the following statements correctly describes a measure of center?
The mean and median can be used to summarize any quantitative data.
The mean is unaffected by extreme values in a small data set.
The median is calculated by adding all of the values in a data set and then dividing by the total number of values.
There can only be one mode in any given set of data.
The median can be used to describe any qualitative data.
Recall that for data that is quantitative, if the dataset is finite there is always a defined mean and median. There is not always a mode.
Measures of Center
14
Which of the following statements correctly describes the variance of a data set?
The variance has the same units as the standard deviation.
The variance is calculated using the median.
The variance is the square of the standard deviation.
The variance has the same units as the data set.
The variance is the square root of the standard deviation.
In order to go from the variance to the standard deviation (SD), we take the square root. Conversely, to get the variance from the SD we simply square the SD.
Standard Deviation
15
The data below shows the number of text messages received by a group of students in a day.
How many students received 10 to 13 messages?
4
6
9
5
The dot plot shows us how many values are at each point. To find the number of students who received 10 to 13 messages, count the number of x's in at each value: 1 student received 10 messages, 2 students received 11 messages, 2 students received 12 messages, and 4 students received 13 messages.
If we count up the number of x's from 10 and 13 values, we see that there are 9 x's, or 9 students.
Dot Plots
16
Using the box-and-whisker plot, match each description with the correct value.
First Quartile
Second Quartile
Third Quartile
A.
52
B.
70
C.
33
D.
29
E.
40
Recall the box shows us Q1, Q2, and Q3. The ends of the box are Q1 and Q3 with the lowest edge (33) being Q1 and the highest (52) being Q3. The line in the box is Q2 or the median, which is 40.
Five Number Summary and Boxplots
17
Jenova has scored ten standardized tests with scores of 65, 88, 46, 72, 77, 90, 95, 59, 66, and 83.
The standardized test score that represents the sixtieth percentile is __________.
66
59
65
77
If we note that there are 10 values, so the 60th percentile can be found with the following calculation:
This tells us that we need to find the 6th ordered value.
The 6th value is 77.
Percentiles
18
Which of the following statements is NOT true about the normal distribution?
The normal distribution is symmetric about the mean.
A large portion of the data is located near the center in a normal distribution.
The normal distribution can be described as “bell-shaped.”
The normal distribution is an example of a bimodal distribution.
We only need to know the mean and standard deviation in order to completely describe a normal distribution.
The normal distribution is a bell-shaped symmetic distribution with only one peak. So it is not bi-modal (i.e. 2 modes), but is unimodal, which means it has 1 mode.
Normal Distribution
19
An outlier is which of the following?
Any value in a data set that is larger than twice the mean value
A value in a data set that is the highest or lowest of the values in the data set
A value in a data set that is significantly higher or lower than most of the values in the data set
A value in a data set that is only significantly lower than most of the values in the data set
Any value in a data set that is larger than twice the median value
An outlier is data that doesn't fit with other data. It is either much larger or much smaller than the other data.
Outliers and Modified Boxplots
20
Rick is an engineer testing the stress required to break samples of steel. He measured the failure stress of 50 samples and found the mean failure stress to be 350 MPa, with a standard deviation of 25 MPa.
If the distribution is normal, the percentage of the data that lies within two standard deviations of the mean is approximately __________.
99.7%
5%
95%
68%
The normal distribution follows the empirical rule, which tells us that within 2 standard deviations we should find 95% of the data.
68-95-99.7 Rule
21
Matt just received his test back. He scored a 78 out of a possible 90 points. His teacher told him the mean score on the test was a 70, with a standard deviation of 5.
Matt’s z-score for the test was __________.
2.4
-2.4
1.6
-1.6
Recall that the z-score can be calculated with the following formula:
The given value is 78 points, the mean is 70 points, and the standard deviation is 5 points. Plug these values in to get the following z-score:
This also tells us that 78 is 1,6 z-scores or standard deviations above the mean.
Standard Scores and Z-Scores
22
Katherine, Jonathan, and Ryan are very competitive friends who went bowling. Afterwards, two of them decided to make bar graphs to plot their scores.
Who do you think made Graph 1 and why?
Jonathan, because he wanted to make the scores appear reasonably close.
Ryan, because he wanted to accurately show each person’s score.
Katherine, because she wanted to make the scores appear reasonably close.
Jonathan, because he wanted to make the scores appear very different.
Katherine, because she wanted to make the scores appear very different.
Although we cannot know for sure, it looks like the bar that is most different between the 2 is for Katherine. She has a much larger value in graph 1. Since they are competitive, it would be logical to assume she overestimated her score to look the best and thus created graph 1.
Misleading Graphical Displays
23
Determine if each graph is positively skewed, negatively skewed, or symmetrical.
= Correct Answer
= Incorrect Answer
Positively (Right) Skewed Negatively (Left) Skewed Symmetrical Distribution
Recall that skew tells us the direction of the tail. So a tail to the right is right skewed, while a tail to the left implies left skewed. If the graph is the same on both sides, we refer to it as symmetric. Finally, a graph with 2 peaks is bimodal.
Shapes of Distribution
24
Consider the times (in seconds) that it took children and adults to solve a Rubik’s cube at a competition.
What does the circled section represent?
It took 7 children 12 seconds to solve the Rubik’s cube.
It took 7 children 21 seconds to complete the Rubik’s cube.
It took 21 children 7 seconds to solve the Rubik’s cube.
Two children took over 70 seconds to solve the Rubik’s cube.
Recall, that a stem and leaf shows the data in stem and leaf form. So, the stem of 7 implies that this is in the tens, so 70 seconds. So the 1 and 2 mean 70+1 or 71 seconds and 70+2 = 72 seconds. This means we can say 2 children took a bit over 70 seconds to complete the cube.
Stem-and-Leaf Plots
25
Which of the following two statements are true?
The range is found by subtracting the minimum value from the maximum value.
The interquartile range is better than the standard deviation to describe skewed data sets.
The interquartile range covers the middle 75% of the data set.
The range is found by subtracting the maximum value from the minimum value.
The definition of the range is simply the max - min value, which is subtracting the min from the max.
Range and Interquartile Range (IQR)
26
In a poll of 216 voters, 134 said they would vote for the candidate from Party X, 52 said they would vote for the candidate from Party Y, and 30 said they would vote for the candidate from Party Z.
If a pie chart were to be made showing the support for each candidate, the smallest central angle would be ________ degrees.
87
50
30
52
Recall that to get the angle for something in a pie chart we use the following formula:
So in this case, the smallest central angle will be associated with the candidate with the least about of votes, which would be from 30 votes for Party Z. The central angle for the candidate from Party Z would be:
So the smallest angle would be 50 degrees.
1
The dotplot below shows the number of text messages received by a group of students in a day.
How many students received less than 15 messages?
15
20
4
9
If we sum up the X's that represent an individual receiving less than 15 messages, we need to include the number of students who received 14 messages, 13 messages, 12 messages, etc. This looks like:
So there are 15 students who received less than 15 messages.
Dot Plots
2
Select the statement that correctly describes a normal distribution.
It is a positively skewed distribution, as the extreme values are greater than the median.
It is a symmetric distribution, as the mean and the median are the same.
It is a negatively skewed distribution, as the extreme values are less than the median.
It is a uniform distribution, as all of the values have equal frequency.
A normal distribution is a bell-shaped and symmetric distribution. So it has a smooth peak, which tells us the mean and median are the same.
Normal Distribution
3
In which of these cases should the median be used?
When the data has extreme values
When the data has nominal values
When the data has small variance
When data has no outliers
Since the mean uses the actual values in the data, it is most affected by outliers and skewness. So, we only want to use the mean when the data is symmetric as a measure of centrality. When the data is skewed or has extreme values,
the median is a better measure since it is not as sensitive to these values.
Measures of Center
4
Let x stand for the percentage of an individual student's math test score. 64 students were sampled at a time. The population mean is 78 percent and the population standard deviation is 14 percent.
What is the standard deviation of the sampling distribution of sample means?
64
1.75
0.22
14
The standard deviation of the sampling distribution is
Center and Variation of a Sampling Distribution
5
The formula for standard deviation of a sample is:
Find the standard deviation of the following data set that has a mean of 6.75:
4, 6, 7, 10
Answer choices are rounded to the hundredths place.
2.17
6.50
2.50
6.25
If we first note the mean of the data is 27/4 = 6.75, we can then get the variance of the data and note that it is:
If we note that the standard deviation(SD) is simply the square root of the variance, then the SD =
Standard Deviation
6
The first quartile (Q1) value from the above box plot is __________.
29
52
40
33
Note the value for Q1 is the left edge of the box, which is 33.
Five Number Summary and Boxplots
7
In a poll of 300 preschoolers, 125 said they preferred chocolate ice cream, 71 said they preferred vanilla, 100 said they preferred cookies & cream, and 4 said they had never eaten ice cream.
If a pie chart were to be made showing the preference for each flavor, the central angle for the chocolate ice cream sector would be __________.
124°
5°
150°
41°
Recall that to get the angle for something in a pie chart we use the following formula:
So in this case, the central angle for the chocolate ice cream sector would be:
Bar Graphs and Pie Charts
8
Which of the following statements is true?
The Central Limit Theorem is applicable only for data sets comprising thirty or more samples.
The Central Limit Theorem is applicable only for data sets comprising less than thirty samples.
The Central Limit Theorem is applicable only for data sets comprising more than thirty samples.
The Central Limit Theorem is applicable only for data sets comprising exactly thirty samples.
Recall that the Central Limit Theorem outlines that when the sample size is large, then the distribution of sample means will be approximately normal. For most distributions, the sample size is n ≥ 30, meaning a sample of 30 or more observations is considered a good sample size.
Shape of a Sampling Distribution
9
For a class reading competition, the students were asked to read a book. Mike, Jack, and Rayon discussed the numbers of pages they read on the first day. One of them made a graph to represent the data.
Who made the graph, and why?
Mike, because he wanted to accurately show the amount read by each person.
Mike, because he wanted to make the amount read by each person appear reasonably close.
Rayon, because he wanted to make the amount read by each person appear very different.
Jack, because he wanted to make it look like he read significantly more than the others.
Since there was a competition, the person who most likely made this graph would want to represent themselves favorably. Since Jack has the most pages, it would probably be him.
Misleading Graphical Displays
10
A baseball scout recorded the type of pitch a pitcher threw during a game and whether it was thrown for a strike or a ball.
Which of the following statements about the stack plot is true?
The pitcher threw the curveball most often.
The pitcher's least accurate pitch (lowest percentage of strikes) is the slider.
The pitcher threw the changeup the most often.
The pitcher's least accurate pitch (lowest percentage of strikes) is the curveball.
For curveballs, it seems like roughly 50% are thrown for a strike. In the other pitches, the percentage of pitches thrown for strikes all seem to be greater than 50% since the strike area is more than half of the total pitches.
Stack Plots
11
Which of the following statements about a positively skewed distribution is true?
The distribution of the data features two modes.
The mean, median, and mode have the same values.
The distribution of the data tails to the right of the median.
The distribution of the data tails to the left of the median.
Skewness refers to how the data trends to the left or right. If a dataset is skewed, it is not symmetric. The direction of the tail of a distribution tells you which direction the skew lies. If there is positive skew, this implies the skew is to the right. If the distribution trends to the right, it will have a mean that is larger than the median due to those higher values.
Shapes of Distribution
12
Consider the histogram showing the heights of individuals on a basketball team.
How many players are taller than 78 inches?
4
12
6
3
If we sum up the frequencies for heights of people who are taller than 78 inches, the totals are:
So there are 12 people taller than 78 inches.
Histograms
13
The midterm exam scores obtained by boys and girls in a class are listed in the table below.
What does the circled section represent?
Eight boys scored over 10 marks on the exam.
Eight boys scored 12 marks on the exam.
Two boys scored between 80 and 89 marks on the exam.
Twelve boys scored 8 marks on the exam.
If we recall that the stem and leaf can give us the actual values in the data set, then the circle corresponds to 81 and 82. We can then note that there are two scores from boys between 80 and 89.
Stem-and-Leaf Plots
14
Mike's electronics store sold 12, 10, 9, 5, 14, 10, and 10 cell phones on each of the seven days of a week.
The mean number of cell phones sold by the store for the week was __________.
9 phones
5 phones
10 phones
70 phones
To get the median we first order the data and take the middle value. The ordered values are:
Since there are an odd number (n=7) of values we simply take the middle, which is 10.
Mean, Median, and Mode
15
In a survey to rate the pizzas served by a pizza parlor, 250 people rated their agreement with the statement, “The pizzas here are one of the best I’ve ever had.” The answers were put into a table.
Rating Frequency
Strongly Agree 27
Agree 50
Neutral 75
Disagree 54
Strongly Disagree 44
The relative frequency of people who strongly agree with the statement is __________.
20%
10.8%
27%
17.6%
To get the relative frequency, we take the frequency of the value and divide it by the total number. So in this case for strongly agree, the relative frequency would be:
Frequency Tables
16
Select the false statement about standard deviation.
It is a measure of how spread out the values of a data set are.
It is calculated using the mean.
It is the square root of variance.
It is the average of the squared differences of the values from the mean.
Recall that the standard deviation (SD) is equal to the square root of the variance and the variance is the average of the squared distances from the mean. So, the standard deviation is the average distance to the mean.
Standard Deviation
17
Ralph records the time it takes for each of his classmates to run around the track one time. As he analyzes the data on the graph, he notices very little variation between his classmates’ times.
Which component of data analysis is Ralph observing?
An outlier in the data set
The overall shape of the data
The center of the data set
The overall spread of the data
Since Ralph is looking at the variation of data, this is examining the spread of the data.
Data Analysis
18
Hannah noted the height of each student in her class and found that the mean height of the students is 56 inches, with a standard deviation of 1.2 inches. The height of one of the students, James, is 59 inches.
What is the z-score for James' height?
2.5
-2.5
-3.6
3.6
Recall that the z-score can be calculated with the following formula:
The given value is 59 inches, the mean is 56 inches, and the standard deviation is 1.2 inches. Plug these values in to get the following z-score:
This also tells us that 59 is 2.5 z-scores or standard deviations above the mean.
Standard Scores and Z-Scores
19
Sara wonders what percentage of her students answered at least half of the quiz questions incorrectly.
The cumulative frequency of students who earned a score of 20 or lower on the quiz is __________.
68
34
54
27
To get cumulative frequency of 20 or less, we simply add up any bin that has the number 20 or less, such as the bin that shows scores of 1-5, 6-10, 11-15, and 16-20.
This would be:
Cumulative Frequency
20
At Brent's school, the final grade for his U.S. History course is weighted as follows:
Tests: 30%
Quizzes: 50%
Homework: 20%
Brent has an average of 82% on his tests, 94% on his quizzes, and 50% on his homework.
What is Brent's weighted average?
74.8%
75.3%
69.6%
81.6%
In order to get the weighted average we use the following formula:
Weighted Mean
21
Jerry graded seven standardized tests with scores of 60, 74, 41, 87, 94, 79, and 57.
Which standardized test score represents the 50th percentile?
41
57
74
79
The 50th percentile will be the median, or middle number. Make sure to first order the data.
The middle number is the 4th value, or 74.
Percentiles
22
Which of the following statements is true?
A standard normal curve has a mean of 0 and a standard deviation of 2.
A standard normal curve has a mean of 1 and a standard deviation of 2.
A standard normal curve has a mean of 0 and a standard deviation of 1.
A standard normal curve has a mean of 1 and a standard deviation of 1.
The standard normal distribution is a normal distribution that is centered at 0 and a standard deviation of 1.
Standard Normal Distribution
23
Naomi weighed 50 patients for a medical study using a scale that measures to the nearest whole pound. She then calculated the mean weight as 176 pounds, with a standard deviation of 12 pounds.
If the distribution is normal, what percent of the data lies between 140 pounds and 212 pounds?
68%
99.7%
34%
95%
Recall that if the data is normal, then the 68-95-99.7 rule applies which states that 68% of all data points fall within one standard deviation of the mean, 95% of all data points fall within two standard deviations of the mean, and 99.7% of all data points fall within three standard deviations of the mean.
140 pounds and 212 pounds are both 36 pounds from the mean of 176 pounds, which is the same as three standard deviations (12 pounds * 3) in either direction. This tells us that 99.7% of the data should lie between 140 cm to 212 cm.
68-95-99.7 Rule
24
A data set has its first and third quartiles as 9 and 17, respectively.
Which of the following data points would be considered an outlier for the data set?
41
27
3
17
To find an outlier we note the lower bound and upper bound for outliers are:
First, find the interquartile range:
Next, plug this value in for IQR, along with Q1 and Q3 to find the lower and upper bounds.
Anything lower than -3 and larger than 29 will be an outlier. 41 is outside of these bounds and is an outlier.
Outliers and Modified Boxplots
25
The graph below shows the change in passenger load factor for all scheduled airlines in the United States over different months of a year.
Which segment of the graph indicates no change in the passenger load?
C
B
D
A
In the graph, when the segment is horizontal, this shows no change in the percentage of passengers. This would indicate no change in passenger load. This is segment B.
Line Charts and Time-Series Diagrams
26
Select the statement that is TRUE.
The interquartile range covers 100% of a data set.
The interquartile range is calculated by adding the first quartile with the third quartile.
The interquartile range is the average value of a data set.
The interquartile range is calculated by subtracting the first quartile from the third quartile.
Recall that the interquartile range is the difference between Q3 and Q1 and can be calculated by subtracting the first quartile from the third quartile:
Range and Interquartile Range (IQR)