STAT*2060DE F
Questions for Assignment #6
This will be another D2L quiz. Please input your responses before the deadline. Give at
least 3 decimal places where applicable.
Questions #1 – #8 refer to the following information.
Marketing organizations sometimes use a pupillometer, a device that used to observe
changes in pupil dilations as the eye is exposed to different visual stimuli, to help them
evaluate potential consumer interest in mew products, alternative package designs, and
other factors. Suppose one organization used a pupillometer to evaluate consumer reaction
to different silverware patterns for a client. 10 consumers were chosen at random, and each
was shown two silverware patterns. The pupillometer readings (in millimeters) are given
in the table below.
Consumer
1
2
3
4
5
6
7
8
9
10
Pattern A
1.12
0.97
0.87
0.97
1.01
1.01
0.88
1.41
1.18
1.08
Pattern B
1.21
1.04
1.05
1.11
1.08
1.15
1.02
1.07
1.20
1.14
N.B. We should NOT be using the independent samples method.
For the following questions, take the differences using Pattern A Pattern B.
#1. What is the point estimate of the population mean difference? [Hint: This value
should be negative]
#2. What is the standard error of the sample mean difference?
#3. What is the 95% confidence interval for the population mean difference?
A) (-.183, .069)
B) (-.173, .059)
C) (-.163, .049)
D) (-.153, .039)
E) (-.143, .029)
Carry out a test that the population mean difference is equal to 0, against a two-sided
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alternative.
#4.What are the appropriate hypotheses?
A) H0 : µ = 0, Ha : µ > 0
B) H0 : µ = 0, Ha : µ = 0
C) H0 : µ1 = µ2 , Ha : µ1 > µ2
D) H0 : p1 = p2 , Ha : p1 = p2
#5. What is the value of the appropriate test statistic? [Hint: This value should be
negative.]
#6. You cannot get an exact p-value from the t table, but you can get an appropriate
range of values. Which one of the following best represents what we can say about the
p-value based on the t table?
A) p-value <.001
B) .025 < p-value <.05
C) .05 < p-value <.10
D) .10 < p-value <.20
E) p-value >.20.
#7. Which one of the following is the most appropriate conclusion at α = .05?
A) There is significant evidence the population mean difference between the two patterns
is equal to 0.
B) There is significant evidence the population mean difference between the two patterns
is not equal to 0.
C) There is not significant evidence that the population mean difference differs from 0.
D) All of the above.
#8. For the above test to be valid, we need the sample differences to be a simple random
sample from population of differences. What is the other assumption that we need? That
is, what else must be true in order for our methods to be reasonable?
A) The population mean difference is known.
B) The population of differences is normally distributed.
C) The two populations have equal variance.
D) The standard deviation of the population of differences is known.
Questions 9-16 refer to the following information:
Suppose a hotel undergoes a renovation, with the notion of catering more toward male
clientele. One year after the renovations have been completed, the hotel conducts a survey
of 24 male and 18 female guests. One question on the survey asks the customers for an
overall satisfaction rating, between 0 and 100. The following table summarizes the results.
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Males
Females
Mean Rating
81.8
62.1
Standard Deviation
7.1
6.8
Sample size
24
18
It is impossible for the ratings to be perfectly normally distributed (one reason is they’re
limited to the 0-100 range). The sample sizes are not very large, but they’re not uncomfortably small either. It’s always useful to plot the data out in boxplots and normal quantile
plots before using our inference procedures.
The boxplots show pretty clear visual evidence of a difference in means. The normal
quantile plots show some deviations from linearity, especially for the women. There are also
some indications of extreme values in the data. When considering using the t procedures,
this should give us some pause. But overall it’s not terrible, so let’s go ahead and use the
t procedures here. The next question is whether we should use the Welch procedure or the
pooled variance version. Since the sample standard deviations are close (7.1 vs 6.8), let’s
go with the pooled variance procedure.
Use the pooled variance t procedure for the following problems. Do NOT use
any large sample approximation.
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#9. What is the value of the pooled sample variance?
#10. What is the value of the standard error of the difference between the sample means?
#11. Construct a 95% confidence interval for the difference in population means
(male-females). Which one of the following is the appropriate interval?
A) 19.7 ± 4.34
B) 19.7 ± 4.39
C) 19.7 ± 4.44
D) 19.7 ± 4.49
E) 19.7 ± 4.54
#12. Test whether the population mean ratings for men and women are equal. Although
one could make an argument for using a one-sided alternative that the male ratings will be
higher than the female ratings on average, use a two-sided alternative. What are the
appropriate hypotheses?
A) H0 : xM
¯
B) H0 : xM
¯
C) H0 : µM
D) H0 : µM
E) H0 : µM
= xF ,
¯
= xF ,
¯
= µF ,
= µF ,
= µF ,
H a : xM = xF
¯
¯
H a : x M < xF
¯
¯
H a : µM = µF
Ha : µM > µF
Ha : µM < µF
#13. What is the value of the appropriate test statistic?
#14. What is the p-value of the test? We cannot get an exact p-value from the table, but
we can find an appropriate range of values. Which one of the following is the appropriate
range?
A) p-value <.001
B) .001 < p-value <.002
C) .01 < p-value <.02
D) .05 < p-value <.10
E) .1< p-value
#15. Is there significant evidence of a difference in population means between men and
women at α = .01?
A)Yes.
B)No.
#16. Which one of the following is the most appropriate conclusion for this hypothesis
test?
A) We can be certain that male guests tend to give higher ratings than female guests at
this hotel.
B) There is very strong evidence that the population mean rating for male guests at this
hotel is greater than the population mean rating for female guests.
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C) There is very strong evidence that the population mean rating for male guests at this
hotel is less than the population mean rating for female guests.
D) There is some evidence that the sample mean rating for male guests at this hotel is
greater than the sample mean rating for female guests at this hotel.
The next series of questions refer to the following information.
Suppose an insurance company is investigating the use of text messsaging by drivers in a
region. As one variable in the study, they are interested in the proportion different age
groups who have texted while driving. They conduct a large survey of drivers in the region,
and find that:
In a random sample of 600 drivers aged 16-19, 372 admitted to texting while driving.
In a random sample of 1500 drivers aged 20-29, 745 admitted to texting while driving.
#17. Calculate a 95% confidence interval for the difference in population proportions
(teenage – twenties). Which one of the following is the appropriate interval?
A) 0.1233 ± 0.046
B) 0.1233 ± 0.048
C) 0.1233 ± 0.050
D) 0.1233 ± 0.052
E) 0.1233 ± 0.054
Test the null hypothesis that the population proportions of drivers who admit to texting
while driving are equal for the two groups. Use a two-sided alternative.
**Please call the 16-19 group Group 1. Your test statistic should be positive**
#18. What are the appropriate hypotheses?
A) H0 : pteen = ptwenties ,
B) H0 : pteen = ptwenties ,
ˆ
ˆ
C) H0 : pteen = ptwenties ,
D) H0 : pteen = ptwenties ,
ˆ
ˆ
Ha : pteen > ptwenties
Ha : pteen > ptwenties
ˆ
ˆ
Ha : pteen = ptwenties
Ha : pteen = ptwenties
ˆ
ˆ
#19. What is the value of the test statistic?
#20. The p-value of the test is closest to which one of the following?
A) .01
B) .05
C) .10
D) .20
E) .50
#21. Which one of the following is the most appropriate conclusion for α = .05?
A) There is significant evidence that the population proportion for the 16-19 age group is
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less than that of the 20-29 age group.
B) There is not significant evidence of a difference between the population proportions for
the two age groups.
C) There is significant evidence that the population proportions for the two age groups are
equal.
D) We can be certain that the sample proportions for the two age groups are equal.
E) There is significant evidence that the population proportion for the 16-19 age group is
greater than that of the 20-29 age group.
#22. Suppose we run a hypothesis test and obtain a p-value of .03. Which one of the
following statements is true?
A) The null hypothesis would be rejected at α = .10, but not at α = .05.
B) The null hypothesis would be rejected at α = .05, but not at α = .01.
C) The null hypothesis would be rejected at α = .01.
D) The null hypothesis would not be rejected at any value of α.
Consider the following output for a two-sample inference procedure for the difference between population means. The output for both the pooled-variance procedure and Welch’s
procedure are given.
Welch Two Sample t-test
data: temp1 and temp2
t = -0.3746, df = 52.842, p-value = 0.7094
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-3.699859 2.535360
sample estimates:
mean of x mean of y
19.54801 20.13026
Pooled Variance Two Sample t-test
data: temp1 and temp2
t = -0.3206, df = 53, p-value = 0.7498
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-4.224599 3.060100
sample estimates:
mean of x mean of y
19.54801 20.13026
#23. What is the p-value of the test that requires the assumption that the population
variances are equal?
#24. Which of these procedures assumes that we are sampling from normally distributed
populations?
A) Just the pooled-variance t procedure.
B) Just the Welch procedure.
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C) Both procedures.
D) Neither procedure.
#25. Which would be the more appropriate procedure in this case?
A) The pooled variance procedure.
B) Welch’s procedure.
C) Impossible to determine with the given information.
#26. Suppose we had 40 pairs of observations of A and B measurements, that resulted in
the following output
Paired t-test
t = -0.3476, df = 39, p-value = 0.73
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-0.001649301 0.001165598
sample estimates:
mean of the differences
-0.0002418514
The differences were taken to be: A readings – B readings.
At α = .05, which one of the following is the most appropriate conclusion?
A) There is significant evidence that the readings are equal on average.
B) There is significant evidence that the A readings are greater than the B readings on
average.
C) There is significant evidence that the A readings are less than the B readings on average.
D) There is not significant evidence of a difference in the readings.
E) We can be certain that on average there is a difference in the A and B readings.
#27. The hydrostatic measurements are less time consuming, and therefore preferable if
they provide reasonable readings when compared to the more accurate but time-intensive
hydrometer readings. The winery will use the hydrostatic measurements if it can be demonstrated that the mean difference between the density measurements of the two measurements does not exceed .002. Looking at the appropriate values in the output, is there
strong evidence that the mean difference in density measurements exceeds .002?
Is there strong evidence that the mean difference in measurements exceeds .002?
A) Yes, there is strong evidence that the mean difference exceeds .002.
B) No, there is not strong evidence that the mean difference exceeds .002.
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