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STAT 3011 Homework Assignment #5
Question
Need help with problem
2(13,14,15,16,18)
STAT 3011 Homework
Assignment #5
Problem II. Additional problems!
13. [2 points] In this problem, you will analyze a dataset. The data are otained from
an experiment where 24 animals were assigned to one of four diets and then blood
coagulation times were measured for each animal. These data are available at
http://users.stat.umn.edu/?guxxx192/courses/data/blood.txt. The response
is called coag and the factor is called diet, with 4 levels.
a. Produce side-by-side boxplots of the response for each level of the factor. Does there
seem to be any strong visual evidence that it is unreasonable to assume the 4 group
distributions of the response have the same standard deviation ??
b. Report the observed sample mean and observed sample standard deviation of the
response for each level of the factor.
c. Carry out the hypothesis test where the null hypothesis is that all group mean blood
coagulation times are equal versus the alternative that at least one is different at the 5%
significance level.
i. What assumptions must be made? State H0 and H a .
ii. Specify the observed test statistic and p-value.
iii. What is the distribution of the test statistic when H0 is true. State the values for its
parameters.
iv. State the conclusion to this test. Based on this conclusion, what type of error could
you be making?
v. Report Tukey HSD simultaneous 95% confidence intervals for the pairwise differences between the group means of blood coagulation times. Interpret this set of
intervals and mention which pairs of group means are statistically significantly
different at the simultaneous 5% significance level.
14. [3 points] An experiment was conducted where paper brightness (measured by a reflectance
meter) was recorded for paper samples produced using 4 different settings (a, b, c, d). Here
is a summary of the dataset:
brightness
setting
Min.
:59.80
a:5
1st Qu.:60.00
b:5
Median :60.50
c:5
Mean
:60.40
d:5
3rd Qu.:60.73
Max. :61.00
We wish to test:
H0 ? µ1 = µ2 = µ3 = µ4
H a ? at least one µ j is not equal to the others
at the 1% significance level, where µ j is the mean brightness for the subpopulation of paper
produced in the jth setting, j = 1, 2, 3, 4. Using R, an ANOVA table was computed:
setting
Residuals
Df Sum Sq Mean Sq F value Pr(>F)
3 1.34000 0.44667 4.2039 0.02261 *
16 1.70000 0.10625
Stat 3011
Homework 5 - Page 3 of 4
Summer 2016
a. What must we assume about these data to conduct this hypothesis test?
b. Using the output above, report the observed test statistic. What is the distribution of
the test statistic when H0 is true? State the values for its parameters.
c. Report the p-value for this test and state the conclusion.
d. If H0 is true, and we planned to independently repeat this experiment, what is the
probability of observing a new p-value that is greater than our current p-value?
15. [2 points] Market researchers know that background music can influence the mood and
purchasing behavior of customers. One study in a supermarket in Northern Ireland compared three treatments: no music, French accordion music, and Italian string music. Under
each condition, the researchers recorded the numbers of bottles of French, Italian, and other
wine purchased. Here is the two-way table that summarizes the data:
Music
Wine
None
French
Italian
Total
French
Italian
Other
30
11
43
39
1
35
30
19
35
99
31
113
Total
84
75
84
243
Perform the chi-square test for this two-way table to see the relationship between music
and wine sales. Give the test statistic, degrees of freedom, the p-value, and your conclusion.
16. [2 points] The following is the R output from a linear regression of poverty rate on high
school graduation rate for the 50 states. (Note. A state in which 10% of the residents are
living below the poverty line has a poverty rate of 10%. Similarly, a state in which 90% of
residents graduate from high school has a high school graduation of 90%.)
> fit <- lm(poverty ~ HighSchool)
> summary(fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 51.69292
7.07829
7.303 2.53e-09 ***
HighSchool -0.47089
0.08272
XXXXX XXXXXXXX
--Residual standard error: 2.293 on 48 degrees of freedom
Multiple R-squared: 0.403,
Adjusted R-squared: 0.3906
F-statistic: 32.41 on 1 and 48 DF, p-value: 7.361e-07
a. Based on the R output, what relationship do you observe between poverty rate and
high school graduation rate?
b. Give the least squares regression line for this relationship. Make sure to correctly label
the response and explanatory variables.
c. Set up the appropriate hypothesis H0 and H a for testing for a significant linear association between poverty rate and high school graduation rate. State a p-value for the test
and use this to make a conclusion at the 0.05 level. (Make sure to state your conclusion
in the terms of this problem!)
Stat 3011
Homework 5 - Page 4 of 4
Summer 2016
Problem III. Bonus problems (Not required to hand in)!
17. [2 points] Goodness-of-fit test. Textbook problem 11.22 (page 555)
18. [2 points] Some believe that individuals with a constant sense of time urgency (often called
type-A behavior) are more susceptible to heart disease than are more relaxed individuals.
Although most studies of this issue have focused on individuals, some psychologists have
investigated geographical areas. They considered the relationship of city-wide heart disease
rates and general measures of the pace of life in the city.
For each region of the United States (Noetheast, Midwest, South, and West) they selected
three large metropolitan areas, three medium-size cities, and three smaller cities. The data
is available at http://users.stat.umn.edu/?guxxx192/courses/data/heart.txt.
In each city they measured three indicators of the pace of life. The variable walk is the
walking speed of pedestrians over a distance of 60 feet during business hours on a clear
summer day along a main downtown street. Bank is the average time a sample of bank
clerks takes to make change for two $20 bills or to give $20 bills for change. The variable
talk was obtained by recording responses of postal clerks explaining the difference between
regular, certified, and insured mail and by dividing the total number of syllables by the time
of their response. The researchers also obtained the age-adjusted death rates from ischemic
heart disease (a decreased flow of blood to the heart) for each city (heart). (Data from R.
V. Levine, “The pace of life.” American Scientist 78 (1990): 450-9.) The variables have been
standardized, so there are no units of measurement involved.
a. Draw a matrix of scatterplots of the four variables.
b. Obtain the least squares fit to the linear regression of heart on bank, walk, and talk.
c. Plot the residuals versus the fitted values. Is there evidence that the variance of the
residuals increases with increasing fitted values or that there are any outliers?
d. Report a summary of the least squares fit. Write down the estimated equation with
standard errors below each estimated coefficient.
References
Agresti, A. and Franklin, C. (2012). Statistics: the art and science of learning from data. Pearson,
3rd ed.
13. [2 points] In this problem, you will analyze a dataset. The data are otained from
an experiment where 24 animals were assigned to one of four diets and then blood
coagulation times were measured for each animal. These data are available at
http://users.stat.umn.edu/?guxxx192/courses/data/blood.txt. The response
is called coag and the factor is called diet, with 4 levels.
a. Produce side-by-side boxplots of the response for each level of the factor. Does there
seem to be any strong visual evidence that it is unreasonable to assume the 4 group
distributions of the response have the same standard deviation ??
b. Report the observed sample mean and observed sample standard deviation of the
response for each level of the factor.
c. Carry out the hypothesis test where the null hypothesis is that all group mean blood
coagulation times are equal versus the alternative that at least one is different at the 5%
significance level.
i. What assumptions must be made? State H0 and H a .
ii. Specify the observed test statistic and p-value.
iii. What is the distribution of the test statistic when H0 is true. State the values for its
parameters.
iv. State the conclusion to this test. Based on this conclusion, what type of error could
you be making?
v. Report Tukey HSD simultaneous 95% confidence intervals for the pairwise differences between the group means of blood coagulation times. Interpret this set of
intervals and mention which pairs of group means are statistically significantly
different at the simultaneous 5% significance level.
14. [3 points] An experiment was conducted where paper brightness (measured by a reflectance
meter) was recorded for paper samples produced using 4 different settings (a, b, c, d). Here
is a summary of the dataset:
brightness
setting
Min.
:59.80
a:5
1st Qu.:60.00
b:5
Median :60.50
c:5
Mean
:60.40
d:5
3rd Qu.:60.73
Max. :61.00
We wish to test:
H0 ? µ1 = µ2 = µ3 = µ4
H a ? at least one µ j is not equal to the others
at the 1% significance level, where µ j is the mean brightness for the subpopulation of paper
produced in the jth setting, j = 1, 2, 3, 4. Using R, an ANOVA table was computed:
setting
Residuals
Df Sum Sq Mean Sq F value Pr(>F)
3 1.34000 0.44667 4.2039 0.02261 *
16 1.70000 0.10625
Stat 3011
Homework 5 - Page 3 of 4
Summer 2016
a. What must we assume about these data to conduct this hypothesis test?
b. Using the output above, report the observed test statistic. What is the distribution of
the test statistic when H0 is true? State the values for its parameters.
c. Report the p-value for this test and state the conclusion.
d. If H0 is true, and we planned to independently repeat this experiment, what is the
probability of observing a new p-value that is greater than our current p-value?
15. [2 points] Market researchers know that background music can influence the mood and
purchasing behavior of customers. One study in a supermarket in Northern Ireland compared three treatments: no music, French accordion music, and Italian string music. Under
each condition, the researchers recorded the numbers of bottles of French, Italian, and other
wine purchased. Here is the two-way table that summarizes the data:
Music
Wine
None
French
Italian
Total
French
Italian
Other
30
11
43
39
1
35
30
19
35
99
31
113
Total
84
75
84
243
Perform the chi-square test for this two-way table to see the relationship between music
and wine sales. Give the test statistic, degrees of freedom, the p-value, and your conclusion.
16. [2 points] The following is the R output from a linear regression of poverty rate on high
school graduation rate for the 50 states. (Note. A state in which 10% of the residents are
living below the poverty line has a poverty rate of 10%. Similarly, a state in which 90% of
residents graduate from high school has a high school graduation of 90%.)
> fit <- lm(poverty ~ HighSchool)
> summary(fit)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 51.69292
7.07829
7.303 2.53e-09 ***
HighSchool -0.47089
0.08272
XXXXX XXXXXXXX
--Residual standard error: 2.293 on 48 degrees of freedom
Multiple R-squared: 0.403,
Adjusted R-squared: 0.3906
F-statistic: 32.41 on 1 and 48 DF, p-value: 7.361e-07
a. Based on the R output, what relationship do you observe between poverty rate and
high school graduation rate?
b. Give the least squares regression line for this relationship. Make sure to correctly label
the response and explanatory variables.
c. Set up the appropriate hypothesis H0 and H a for testing for a significant linear association between poverty rate and high school graduation rate. State a p-value for the test
and use this to make a conclusion at the 0.05 level. (Make sure to state your conclusion
in the terms of this problem!)
Stat 3011
Homework 5 - Page 4 of 4
Summer 2016
Problem III. Bonus problems (Not required to hand in)!
17. [2 points] Goodness-of-fit test. Textbook problem 11.22 (page 555)
18. [2 points] Some believe that individuals with a constant sense of time urgency (often called
type-A behavior) are more susceptible to heart disease than are more relaxed individuals.
Although most studies of this issue have focused on individuals, some psychologists have
investigated geographical areas. They considered the relationship of city-wide heart disease
rates and general measures of the pace of life in the city.
For each region of the United States (Noetheast, Midwest, South, and West) they selected
three large metropolitan areas, three medium-size cities, and three smaller cities. The data
is available at http://users.stat.umn.edu/?guxxx192/courses/data/heart.txt.
In each city they measured three indicators of the pace of life. The variable walk is the
walking speed of pedestrians over a distance of 60 feet during business hours on a clear
summer day along a main downtown street. Bank is the average time a sample of bank
clerks takes to make change for two $20 bills or to give $20 bills for change. The variable
talk was obtained by recording responses of postal clerks explaining the difference between
regular, certified, and insured mail and by dividing the total number of syllables by the time
of their response. The researchers also obtained the age-adjusted death rates from ischemic
heart disease (a decreased flow of blood to the heart) for each city (heart). (Data from R.
V. Levine, “The pace of life.” American Scientist 78 (1990): 450-9.) The variables have been
standardized, so there are no units of measurement involved.
a. Draw a matrix of scatterplots of the four variables.
b. Obtain the least squares fit to the linear regression of heart on bank, walk, and talk.
c. Plot the residuals versus the fitted values. Is there evidence that the variance of the
residuals increases with increasing fitted values or that there are any outliers?
d. Report a summary of the least squares fit. Write down the estimated equation with
standard errors below each estimated coefficient.
References
Agresti, A. and Franklin, C. (2012). Statistics: the art and science of learning from data. Pearson,
3rd ed.
|
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