Stats

STAT A410/A610

Exam 1

READ THE INSTRUCTIONS FIRST

• Do Not Discuss the Exam with Others

• Use blank white papers and a black ballpoint pen to write your solutions to the problems. You may

use a WORD document to type in the answers.

Be sure to copy the outputs from R including

graphs and paste them on the WORD document.

• Write your name on the first page of your answer sheets.

• Please show equations used and calculations for each problem to get full credit.

• Your completed solution sheets are due by 5:00 pm on Monday, September 21.

• Scan your answer sheets as one PDF document. Go to Tests on Blackboard and upload your PDF

document.

STAT A410/A610

Statistical Methods/Advanced Statistical Methods

Fall 2020

EXAM 1

(Upload to Blackboard by 5:00pm, Monday, Sep 21)

Show the scripts and the output for any calculation using

UNLESS OTHERWISE SPECIFIED, USE  = 0 05. , LEVEL OF SIGNIFICANCE.

DO NOT DISCUSS THE EXAM PROBLEMS WITH

OTHERS

Problem 1

Many boxed cake mixes include special high-altitude baking instructions. To determine

any difference between baking times at low and high altitudes, the consumer group

Public citizen made several similar cakes in nine-inch rounded pans in Miami and

Denver, and carefully recorded the time to bake (in minutes). The summary statistics for

the data is given as follows:

(a) Describe each box plot in terms of

center, shape, spread, and outliers.

(b) Describe the similarities and differences between the two distributions. (c) State the hypothesis to test for the homogeneity of variances (d) Test the hypotheses in part (c) at 5% level of significance. Obtain the exact p-value

using , and state your conclusion.

(e) State the hypotheses to compare the mean baking times for the two altitudes. (f) Test the hypothesis in part (e) at 1% level of significance using an appropriate test

statistic based on your conclusion in part (d). Obtain the exact p-value using ,

and state your conclusion.

Problem 2

A psychologist rated subjects undergoing withdrawal from narcotics on the basis of the

extent of their depression before and one hour after receiving a dose methadone. Degree

of depression was rated as N (not depressed), M (mildly depressed), or S (severely

depressed). The results are shown in following table. One would like to investigate

High

Altitude

Low

Altitude

Sample

Size 14 20

Mean

27.207 24.785

Standard

Deviation 2.692

.951

whether the subjects undergoing withdrawal from narcotics tend to be more depressed

after receiving a dose of methadone than before.

Subject 1 2 3 4 5 6 7 8 9 10

Before N N M N M M M N N N

After M M S M M N S S M M

(a) What is the scale of measurement for the degree of depression? (b) State the hypotheses verbally and probabilistically. (c) Use the appropriate test statistic and determine the p value. (d) What do you conclude about the withdrawal of narcotics on degree of depression

at 5% level of significance? Write a statement explaining your findings.

Problem 3

To examine the growth of cork boring of constant diameter were taken from the northern

and eastern directions of the trunk of 12 trees in a plantation. The weights of these boring

(measured in centigrams) are given in the table below:

Tree no. Northern (N) Eastern (E)

────────────────────────────────────

1 72 66

2 60 53

3 56 57

4 41 29

5 32 32

6 30 35

7 39 39

8 42 43

9 37 40

10 33 29

11 32 30

12 63 45

(a) Are the two samples independent or paired (related)? (b) The above figure represents the Box plot for the differences (N – E). Describe the

distribution of the differences using the Box plot.

(c) State hypotheses to investigate whether the weights of cork borings for the northern and eastern directions of trunks differ.

(d) Obtain the “signed ranks’ for the differences (N - E), and calculate the value of the Z test statistic and the p-value to test the hypotheses in part (c). Be sure to

include the continuity correction (use the formula on the ppt slides; the text book

has the continuity correction in the wrong place.

(e) Perform the nonparametric test using and compare the answers. (f) What do you conclude about the growth of cork boring for the two directions of

the trunks based on the p-value at 5% level of significance?

Problem 4

A horticulturist experimenting with two cross-matches in yews wants to develop a strain

with similar means in height. The following are sample heights of the two cross-matches:

Cross-match A Cross-match B

1.58 2.08 2.36 2.09 2.37 1.91 2.06 2.19 1.91 1.52 2.31 1.22

2.27 1.62 1.89 2.00 1.39 2.00 2.10 1.44 2.20 2.57

The combined ordered data is given below:

1.22 1.39 1.44 1.52 1.58 1.62 1.89 1.91 1.91 2.00 2.00 2.06 2.08 2.09 2.10

2.19 2.20 2.27 2.31 2.36 2.37 2.57

(a) Are the two samples independent or related? (b) State the hypotheses. (c) Assign ranks to the combined data. (d) Perform a nonparametric method to test the hypothesis. Be sure to include the

continuity correction. State the p-value.

(e) Perform the nonparametric test using and compare that to the answer in part (d).

(f) What do you conclude about the two cross-matches with regard to heights of yews?

Problem 5

Identify the experimental units, the factors, the treatments, and the response variables for

the following examples:

(a) Ability to grow in shade may help pines found in the dry forests of Arizona to resist drought. How well do pines grow in shade? Investigators planted pine

seedlings in a greenhouse in either full light, light reduced to 25% of normal by

shade cloth, or light reduced to 5% of normal. At the end of the study, they dried

the young trees and weighed them.

(b) Sickle-cell disease is an inherited disorder of the red blood cells that in the United States affect mostly blacks. It can cause severe pain and many complications.

Can the drug hydroxyurea reduce the severe pain caused by sickle-cell disease?

A study by the National Institutes of Health gave the drug to 150 sickle-cell

sufferers and a placebo to another 150. The researchers then counted the episodes

of pain reported by each subject.

(c) Is diet or exercise effective in combating insomnia? Some believe that cutting out desserts can help alleviate the problem, while others recommend exercise. Forty

volunteers suffering from insomnia agreed to participate in a month-long

experiment. Half were randomly assigned to a special no-desserts diet; the others

continued desserts as usual. Half of the people in each of these groups were

randomly assigned to an exercise program, while the others did not exercise.

Those who ate no desserts and engaged in exercise showed the most

improvement.

(d) Some schools teach reading using phonics (the sounds made by letters) and other using whole language (word recognition). Suppose a school district want to know

which method works better. They use the PACT scored to determine reading

ability.

(e) An experiment is conducted to compare a new variety of corn called opaque-2 with normal corn as food for chicks. The researchers decide to serve each of the

two types of corn at two protein levels: 15% protein and 20% protein. They feed

each diet to 10 one- day-old chicks and record their weight gains after 21 days.

(f) Doctors investigated the relationship between a person’s heart rate and the frequency at which that person stepped up and down on steps of various heights.

There were 3 rates of stepping and 2 different step heights used. A subject

performed the activity (stepping at one of the 3 stepping rates at one of the 2

possible heights) for three minutes. His heart rate was then measured.