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Mathematics 37

Lab 7- Probability Distributions

April 6, 2017

Due On Canvas: 11:00pm April 11, 2017

 

Lab Objectives:

·         Applying the Central Limit Theorem to probability questions.

·         Compute one population confidence intervals for proportions and means

·         Determine the sample size need for a desired margin of error.

 

R-Instructions:

·         One-Population Proportion Confidence Interval

·         One-Population Mean Confidence Interval

·         Normal Distribution Probabilities

·         Any other R-instructions used in  Lab 1-Lab 6  may be needed

Packages Needed for Lab:

·         No new packages

·         ggplot2 may be needed for review problems

 

 

 

Introduction

           

Confidence Intervals are ranges of values in which it is believed that a parameter lies.  The structure of a confidence interval is

 

 

 

The point estimate is the corresponding statistic to the parameter that is being estimated.  The multiplier is associated with the level of confidence.  The larger the multiplier, the wider the interval, resulting in a higher level of confidence that the interval contains the parameter being investigated. The standard error is an estimate of the standard deviation.  Together, the multiplier and the standard error are the margin of error.  The margin of error is the number of standard deviation from the point estimate, in both directions. 

The width of the confidence interval can be controlled by the level of confidence or the sample size.  Often in industry, a standard confidence level is used.  Thus, to control the width of the confidence interval, a statistician will carefully select the sample size. 

 

 

Lab 7 Assignment

Part A. Constructing Confidence Intervals

 

What do you need to submit? 

·         Submit R coding and outputs for all problems. 

·         Complete the table and fill in the blanks.

·         Answer question in complete sentences.

 

 

1.     (16 points) Using the given confidence levels given in the table to construct confidence intervals for the mean furthest distance students have traveled. And complete the table.

 

Confidence Level	Confidence Interval	Margin of Error	Length of Interval
80%
85%
90%
95%

 

·              R-codes: 4 points

·              Answers: 12 points

 

 

2.      (1 point) With all other things fixed, what can you conclude about the relation between the level of confidence and the margin of error?  (1 point)

 

As the level of confidence increase, the margin of error _____________

 

3.      (1 point) With all other things fixed, what can you conclude about the relation between the level of confidence and the length of confidence interval? (1 point)

 

As the level of confidence increase, the length of interval _____________

 

 

4.      (3 points) Interpret the 95% confidence interval for the mean furthest distance students have traveled.

·               R-codes: 1 point

·              Answers:  2 points

 

 

 

 

 

5.      (2 points) What does the “95% level”  mean in Question 4?

 

6.      ( 5 points) Now construct and interpret a 99% confidence interval for the percent of students who admit to being afraid of clowns. 

 

·               R-codes: 1 point

·              Answers:  4 points

 

 

Part B. Determining Sample Size

 

   What do you need to submit? 

·         Submit R codes for all problems. 

·         Neatly organize and label responses. 

 

7.      (4 points)  What sample size is needed to be 92% confident that the proportion of students who fear clowns is estimated to within 4%?  Use as the best guess estimate, the sample proportion from the class dataset. 

 

 

8.      (4 points)  What sample size is needed to be 98% confidence that the mean furthest distance students have traveled is estimated to within 250 miles? 

Note: To estimate the standard deviation, use

 

      

 

from the sample data set. 

 

 

Part C. Review Problems

 

What do you need to submit? 

·         Submit R coding for all problems. 

·         Answer question in complete sentences.

·         Neatly organize and label responses. 

 

 

9.      (3 points) The mean early-bird special admission price for a movie is $7.50.  If the distribution   of a movie admission charges is approximately normal with a standard deviation of $1.25, what is the probability that a randomly selected admission charge is less than $7.25?  (1 point for R-codes and 2 points for answer)

 

 

10.  (6 points)  The national mean SAT score is 1490 and the standard deviation is 315.  Suppose that nothing is known about the distribution shape.  If a random sample of 200 scores were selected:

a.       What is the probability that the sample mean exceeds 1500?  Is this a rare event?  (1 point for R-codes and 2 points for answer)

b.      What is the probability that the sample mean exceeds 1550?  Is this a rare event?  (1 point for R-codes and 2 points for answer)

 

11.  (12 points) A pizza shop owner determines the number of pizzas that are delivered each day. 

 

 

a.       What is the probability that more than 37 pizzas were delivered on any given day? (1 point for R-codes and 1 point for answer)

 

b.      What is the probability that at most 36 pizzas were delivered on any given day? (1 point for R-codes and 1 point for answer)

 

 

c.       Find the mean, the variance, and the standard deviation for the distribution shown.  (2  points  for R-codes and 2 points for answer)

 

d.      Find the interval for  (1 point for R-codes and 1 point for answer)

 

 

e.       What is the probability that the number of pizzas delivered on any given day more than one standard deviation away from the mean? (1 point for R-codes and 1 point for answer)

 

 

 

 

12.  (6 points) Using ggplot2 construct side- by-side boxplots for the dollar amount with which a person will gamble using a grouping variable of fear of ridicule.  Describe any differences or similarities between the two boxplots.

 

·         R-codes for the side-by-side boxplot: 2points

·         Side-by-side boxplots with appropriate label of x-and y-axes, title, and proper size of the plot: 2 points

·         Describing any differences or similarities between the two boxplots: 2 points

 

Total Number of Points=63 points + 2 free Points=65 points