just look at the instruction and use the ggplot 2 to do it

Mathematics 37

Lab 6- Probability Distributions

March 9, 2017

Due On Canvas: 11:00pm March 22, 2017

 

Lab Objectives:

·         Find the mean and standard deviation of a discrete random variable. 

·         Calculate probabilities associated with normal distributions

·         Applying the Central Limit Theorem to probability questions.

R-Instructions:

·         Mean and standard deviation of a discrete random variable

·         Normal Distribution Probabilities

·         Any other R-instructions used in  Lab 1-Lab 4  may be needed

Packages Needed for Lab:

·         No new packages

·         ggplot2 may be needed for review problems

 

Introduction

            In class, we have discussed  probability distributions for both discrete and continuous random variables.  A probability distribution is a distribution showing a random variable and its associated probabilities.  Continuous random variables are variables whose values are the values on an interval.  A discrete random variable is one in which the values are not continuous.

            Probability distributions can be displayed as a graph regardless if it is associated with a discrete or a continuous random variable and can be numerically describe using the mean and the standard deviation.  The mean and variance of a probability distribution for a discrete random variable can be calculated using:

To find the mean and standard deviation of a probability distribution for a continuous random variable, calculus is needed. 

            A specific probability distribution is the standard normal distribution.  The standard normal distribution is a bell-shaped distribution with a mean of  and a standard deviation of   The z-table reports the area under the standard normal distribution.

            The relationship between the standard normal distribution and a normal distribution in the standardize z-score:

This relation is the conversion factor between the two distributions.  The tricky part comes in determining the value of the standard deviation.  Its value is determined by the Central Limit Theorem.

 

Lab 6 Assignment

 

!!!Plagiarism!!!

Although students are encouraged to work together, the responses that a student submits for grading should be his or her own individual efforts.  To submit another student’s work and call it your own is a violation of the Honor Code of the University of the Pacific.

 

Directions:  Use R in all calculations.  Provide coding associated with each problem. 

 

Part A. (12 points) Mean and Standard Deviation of a Probability Distribution of a Discrete Random Variable

 

1.      (  6 points--4 points for codes, and 2 points for answers)Find the mean and the standard deviation of the following probability distribution

 

 

2.      (  6 points--2 points for codes, and 4 points for answers) The maximum patent life for a new drug is 17 years.  Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life of the drug – that is, the length of time a company has to recover research and development costs and make a profit. Suppose the distribution of the lengths of patents life for new drugs is as shown here:

 

 

a.       What is the probability that the patent life for a new drug will exceed 7 years?

b.      What is the expected (mean) number of years of the patent life for a new drug?

c.       What is the standard deviation for the number of years of the patent life for a new drug?

d.      Find the probability that x (years) falls into the interval of .

 

Part B. (10 points) Standard Normal Distribution

3.      (  6 points--3 points for codes, and 3 points for answers) Evaluate

a.      

b.     

c.      

 

4.      (  4 points--2 points for codes, and 2 points for answers) Find the value of b:

a.      

b.     

Part C. (10 points) Probability Calculations Involving Normal Distributions

 

5.      (  2 points—1  point for codes, and 1 point for answer )Evaluate

        where  and

 

6.      (  2 points—1  point for codes, and 1 point for answer ) Suppose the number of a bacteria in samples of 1 milliliter (ml) of drinking water tend to be approximately normally distributed with a mean of 85 and a standard deviation of 9.  What is the probability that a given 1 ml sample will contain more than 100 bacteria?

 

7.      (  4 points—2  points for codes, and 2 points for answers ) Weights of female cats of a certain breed are normally distributed with mean 4.1 kg and standard deviation of 0.6 kg. 

a.       Approximately, what percent of female cats have weights between 3.7 kg and 4.4 kg?

b.      How heavy is a female cat whose weight is on the 80^th percentile (20% of female cats weigh more than this cat)?

 

Part C.  (24 points) Review Problem

8.      The offering price of a house is based on the number of square feet in the house plus many other factors.  Consider the following information collected from a single town. 

 

 

Note: You may enter the data into an Excel file then import into R or create a data frame as you did in Lab 1.

a.       Using ggplot2, construct a scatterplot with linear regression line showing the offering price against the house square footage.  House square footage should be the explanatory variable. (  4 points—2 points for codes, and 2 points for scatterplot )

b.      What is the correlation coefficient between the number of square feet in a house and its offering price? (  2 points—1  point for codes, and 1 point for answer )

c.       What is the linear regression equation between square feet and its offering price in which offering price is the response variable? (  2 points—1  point for codes, and 1 point for answer )

d.      What offering price is predicted if the house has 2000 ft². (  2 points—1  point for codes, and 1 point for answer )

e.       Construct the standardized residual plot for the regression analysis.  Does there appear to be any large residuals?  Is there a pattern to the display?  Discuss. (  6 points—2  point for codes, 2 points for residual plot, and 2 points for discussion )

f.       What is the value of the coefficient of determination?  Write a sentence to interpret this value in terms of the amount of variation. (  3 points—1  point for answer, and 2 points for interpretation )

g.      Considering the variables of square footage of a house and the offering price of a house

                                                              i.      Calculate the coefficient of variation, as percentages, for  the square footage of a house.   (  3 points—2  points for codes, and 1 point for answer )

                                                            ii.      Calculate the coefficient of variation, as percentages, for  the offering price of a house.  (  3 points—2  points for codes, and 1 point for answer )

                                                          iii.      Which variable has more variation with respect to its mean: square footage of a house or the offering price of a house? (  1  point for answer )

 

 

Total Number of Points=56 points + 4 free Points=60 points