STAT412
Homework 7 Due Friday, Nov. 22
Unless you are specifically asked to do the problem by hand, you can use statistical software.
1.
Standardized tests, such as the Comprehensive Test of Basic Skills (CTBS), are used by school systems to evaluate the performance of their students. CTBS scores for two different areas, such as math and reading, can be plotted in a scatterplot for a group of randomly to see if the scores show evidence of a linear relationship.
Construct scatterplot and calculate Pearson correlation coefficient. Interpret the results!
Student Reading Score Math Score
1 47 42
2 71 81
3 64 68
4 35 43
5 43 50
6 60 75
7 38 47
8 59 59
9 67 69
10 56 57
11 67 57
12 57 54
13 69 75
14 38 38
15 54 59
16 76 63
17 53 57
18 40 40
19 47 52
20 23 22
2. The paper "The Relation Between Freely Chosen Meals and Body Habits" (Amer. J. Clinical Nutrition
(1983): 32-40) reported results of an investigation into the relationship between body build and energy intake of an individuals diet. A measure of body build is the Quetelet index (x) with a high value of x indicating a thickset individual. The variable reflecting energy intake is y=dietary energy density. There were nine subjects in the investigation.
Calculate r with and without subject 9 included. What do you learn?
Subject x Y
1 221 .67
2 228 .86
3 223 .78
4 211 .54
5 231 .91
6 215 .44
7 224 .90
8 233 .94
9 268 .93
3.
Underinflated or overinflated tires can increase tire wear and decrease gas mileage. A manufacturer of a new tire tested the tire for wear at different pressures. Construct scatterplot and calculate Pearson correlation coefficient. The researcher for this study calculated a Pearson correlation coefficient and concluded no relationship existed between the two variables. Was he correct in his conclusion?
Pressure Mileage(thous)
30 29.5
30 30.2
31 32.1
31 34.5
32 36.3
32 35
33 38.2
33 37.6
34 37.7
34 36.1
35 33.6
35 34.2
36 26.8
36 27.4
4. The following table provides information on life expectancies for a sample of 22 countries. It also lists the number of people per television set in each country.
Country Life Expectancy People Per TV
Angola 44 200
Australia 76.5 2
Cambodia 49.5 177
Canada 76.5 1.7
China 70 8
Egypt 60.5 15
France 78 2.6
Haiti 53.5 234
Iraq 67 18
Japan 79 1.8
Madagascar 52.5 92
Mexico 72 6.6
Morocco 64.5 21
Pakistan 56.5 73
Russia 69 3.2
South Africa 64 11
Sri Lanka 71.5 28
Uganda 51 191
United Kingdom 76 3
United States 75.5 1.3
Vietnam 65 29
Yemen 50 38
Instructions: When Minitab is used to answer a question below, copy the output from Minitab into your document.
a) Use Minitab to produce a scatterplot of life expectancy vs. people per television set. Does there appear to be an association between the two variables? Elaborate briefly.
b) Have Minitab calculate the value of the Pearson correlation coefficient between life expectancy and people per television.
c) Since the association is so strongly negative, one might conclude that simply sending television sets to the countries with lower life expectancies would cause their inhabitants to live longer. Comment on this argument.
d) If two variables have a correlation close to +1 or –1, indicating a strong linear association between them, does it follow that there must be a cause-and-effect relationship between them?
This example illustrates the very important distinction between association and causation. Two variables may be strongly associated (as measured by the correlation coefficient) without a cause-and-effect relationship existing between them. Often the explanation is that both variables are related to a third variable not being measured; this variable is often called a lurking variable.
e) In the case of life expectancy and television sets, suggest a lurking variable that is associated both with a country’s life expectancy and with the prevalence of televisions in the country.
5. If you peruse the bookshelves of a typical college professor, you will find a variety of books ranging from textbooks to esoteric technical publications to paperback novels. In order to determine whether or not the price of a book can be determined by the number of pages it contains, a college professor recorded the number of pages and price for 15 books on one shelf. The data are shown below.
Pages Price Pages Price Pages Price
104* 32.95 342* 49.95 436 5.95
188* 24.95 378 4.95 458* 60.00
220* 49.95 385 5.99 466* 49.95
264* 79.95 417 4.95 469 5.99
336 4.50 417* 39.75 585 5.95
*Denotes Hardback Book
a. Relating # of pages and price, would you expect a positive or negative correlation?
b. Construct scatterplot and calculate Pearson correlation coefficient. Depict type of book on your scatterplot.
c. Construct scatterplot and calculate Pearson correlation coefficient using just the data for the hardback books.
d. Construct scatterplot and calculate Pearson correlation coefficient using just the data for the paperback books.
e. What do you learn from parts b, c, and d above.
6. Studies have shown that people who suffer sudden cardiac arrest (SCA) have a better chance of survival if a defibrillator is administered very soon after cardiac arrest. How is survival rate related to the time between when cardiac arrest occurs and when the defibrillator shock is delivered? This question is addressed in the paper “Improving Survival from Sudden Cardiac Arrest: The Role of Home Defibrillators” (by J.K. Stross, University of Michigan, February 2002). The accompanying data give y = survival rate (percent) and x = mean call-to-shock time (minutes) for a cardiac rehabilitative center (where cardiac arrests occurred while victims were hospitalized and so the call-to-shock time tended to be short) and for four communities of different sizes.
Mean call-to-shock time,x 2 6 7 9 12
Survival Rate, y 90 45 30 5 2
Do the following by hand and on Minitab.
a. Construct a scatter plot.
b. Calculate the Pearson correlation coefficient.
c. Determine equation of least squares line that can be used for predicting a value of y based on a value of x.
d. Compute SSE =
( y y ˆ )
2for the least squares line.e. Why do we call the least squares line the “best fitting line”?
f. Calculate r2 using the following formula:
2 2 2
2
) (
ˆ) ( ) (
y y
y y y
r y . Interpret the r2 value.
g. Using your equation in part c, draw the least squares line on the scatterplot you constructed in part a.
h. Use your prediction equation to predict SCA survival rate for a community with a mean call-to-shock time of 5 min.
7. Physical Characteristics of sharks are of interest to surfers and scuba divers as well as to marine researchers. Because it is difficult to measure jaw width in living sharks, researchers would like to determine whether it is possible to estimate jaw width from body length, which is more easily measured. The following data on x = length (in feet) and y = jaw width (in inches) for 44 sharks was found in various articles appearing in the magazines Skin Diver and Scuba News:
x y x y x y x y
18.7 17.5 14.6 13.9 16.7 15.2 12.2 14.8
12.3 12.3 15.8 14.7 17.8 18.2 15.2 15.9
18.6 21.8 14.9 15.1 16.2 16.7 14.7 15.3
16.4 17.2 17.6 18.5 12.6 11.6 12.4 11.9
15.7 16.2 12.1 12.0 17.8 17.4 13.2 11.6
18.3 19.9 16.4 13.8 13.8 14.2 15.8 14.3
14.3 13.3 13.6 14.2 16.2 15.7 15.7 14.3
16.6 15.8 15.3 16.9 22.8 21.2 19.7 21.3
9.4 10.2 16.1 16.0 16.8 16.3 18.7 20.8
18.2 19.0 13.5 15.9 13.6 13.0 13.2 12.2
13.2 16.8 19.1 17.9 13.2 13.3 16.8 16.9
Use Minitab to answer the following questions.
a. Construct scatterplot
b. Calculate Pearson correlation coefficient.
c. Determine the equation for the least squares line.
d. Calculate R2 and interpret
e. Conduct test of H0: B1 = 0 vs Ha: B1
0 at =.05.f. Estimate the mean jaw width for sharks of length 15ft using a 95% confidence interval.
g. Assess the reasonableness of the assumptions that are required for parts e and f.
