STAT 35000 University of Indianapolis Statistics Exam Practice

Stat 35000 Exam 1 Spring 2021

Time Limit: 150 minutes

(Including downloading the file, scanning the solution to a single PDF file and submitting it twice to “Exam 1” and “Exam 1 Second Submission” on Canvas)

Your name:

• It is an open-book exam, and you can refer to all course materials.
• You are responsible for upholding IUPUI’s standard for academic integrity. This includes protecting your work from the eyes of other students.
• When you are done, please scan your solution to a SINGLE PDF file and upload it on canvas twice through ”Exam 1” and ”Exam 1 Second Submission”.

•Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.

•Report decimal answers to three decimal places.

•Organize your work, in a reasonably neat and coherent way, in the space provided or your own plain papers. Work scattered all over the page without a clear ordering will receive very little credit.

Problem	PointsPossible	Points Earned
1	18
2	20
3	30
4	20
5	22
6	20
7	20
Total	150
Bonus	15

Problem 1. (5+3+5+5=18 points) The scores, of a random sample of students on an ‘aptitude test’ are:

133 165 165 122 103 126 126 196 148 154 112 136 154 176 137 115 152 100

1. Draw an ordered stem-and-leaf plot (Using the hundreds and tens places as the stems).

2.Without calculation, based on the shape of the stem-and-leaf plot, is median bigger than mean?

3.Compute the five-number summary of the data.

4.Compute the IQR and determine whether there are any outliers. Then draw a horizontal Boxplot.

4.

Problem 2. (5+5+10=20 points) Suppose that there are two candy jars: Jar #1 has 20 chocolates and 30 mints, while Jar #2 has 25 of each. Candice does not prefer a jar or a type of candy. She picks a jar at random, and then picks a candy at random from the selected jar.

• What is the probability that Candice picked Jar #1 and get a mint from the jar?
• What is the probability that Candice picked a mint?
• If the selected candy was found to be a mint, what is the probability that it came from Jar #1?

Problem 3. (3+5+5+7+10=30 points) A contractor is planning to bid on a small construction project and finds that the number of days, , required to complete it, has a discrete distribution with probability mass function given in the following table:

x 10 11 12 13

p(x) 0.15 0.20 0.35 ?

1. What is the probability that it will take him exactly 13 days to complete the project?
2. Given that it will take 11 or more days to complete the project, what is the probability that it will take him exactly 13 days to complete the project?
3. Calculate the expected value of X i.e. E(X).
4. Calculate the variance of X, i.e., V(X).
5. Find V[3X +10].

Problem 4. (5+5+5+5=20 points) Let X be a random variable with probability density function given by

1.What is the value , so that f (x) is a valid PDF?

2.Let , compute .

3.Find the CDF, , of X.

• Compute the probability that X is greater than 0.5, i.e., P (X > 0.5).

.

Problem 5. (5+5+7+5=22 points) It is known that 25% of the students at IUPUI are smokers. Five students are selected at random from the IUPUI student population, and let be the number of smokers among them.

• Identify by name the distribution of the random variable , and specify its parameter values.
• Please give the formula for the PMF of X. Also, remember to give the domain of the distribution!
• Given that there are at least 1 of the five selected students is a smoker, what is the probability that exactly 1 of the five students is a smoker?
• How many smokers do you expect in this group of 5 students?

,

Problem 6. (5+5+10=20 points) The achievement scores for a college entrance examination are normally distributed with mean 75 and standard deviation 10.

1. What fraction of the scores lies between 72 and 82?
2. What is the standardized score (or z-score) of 90 in the achievement score distribution?
3. Find the IQR of the achievement score distribution.

Problem 7. (2+6+4+8=20 points) The Department of Biology at Kenyon College conducted an experiment to study the growth of pine trees at a site located just south of Gambier, Ohio, on a hill overlooking the Kokosing River valley. They want to know how to predict the heights of the pine trees based on their diameters. Height and diameter measurements were taken from 859 random selected pine trees at the site in 1997. Summary statistics of the data are

, , , , ,.

1.In view of the study, what is the independent variable X, and what is the response variable Y?

• Calculate the correlation coefficient between the height and the diameter. Is the linear association positive or negative, and is it strong, moderate or weak?
• What proportion of variation in the heights of the pine trees is explained by their diameters?
• Calculate the least squares estimates of the coefficients a and b of the regression equation, i.e., .

Bonus Problem: (15 points) Given and , what is ?