1. (20.01) A study of commuting times reports the travel times to work of a random sample of 1000 employed adults in Chicago.

The mean is x = 40.0 minutes and the standard deviation is s = 56.9 minutes.
What is the standard error of the mean?

21.88

1.80

6.99

1.09

2. (20.03) Use Table C or software to find the following critical values.

Step 1:
the critical value for a one-sided test with level a = 0.05 based on the t(3) distribution.

3.747

3.143

2.862

2.353

________________________________________

Step 2:
the critical value for a 98% confidence interval based on the t(25) distribution.

1.321

1.282

2.325

2.485

3. (20.04) You have an SRS of size 30 and calculate the one-sample t statistic

Step 1:
What is the critical value t* such that t has probability 0.025 to the right of t*?

-2.064

2.045

-2.060

2.060

________________________________________

Step 2:
What is the critical value t* such that t has probability 0.75 to the left of t*?

0.315

0.683

-0.685

-0.315

4. (20.05) What critical value t* from Table C would you use for a confidence interval for the mean of the population in each of the following situations? (If you have access to software, you can use software to determine the critical values.)

Step 1:
A 95% confidence interval based on n = 12 observations.

2.201

2.521

1.812

1.372

________________________________________

Step 2:
An 99% confidence interval from an SRS of 2 observations.

42.210

72.358

36.353

63.657

________________________________________

Step 3:
A 90% confidence interval from a sample of size 1001.

3.499

1.646

3.355

1.232

5. (20.06) Our decisions depend on how the options are presented to us. Here?s an experiment that illustrates this phenomenon. Tell 20 subjects that they have been given $50 but can?t keep it all. Then present them with a long series of choices between bets they can make with the $50. Scattered among these choices in random order are 64 choices that ask the subject to choose between betting a fixed amount and an all-or-nothing gamble. The odds for all the bets are the same, but in 32 of the choices, the fixed option reads ?Keep $20? and in the other 32 choices the fixed option reads ?Lose $30.? These two fixed options lead to exactly the same outcome, but people are more likely to choose the fixed option that says they lose money. Here are the percent differences (?Number of times chose ?Lose $30?? minus ?Number of times chose ?Keep $20?? divided by the number of trials on which the 20 subjects chose the fixed-option gamble rather than the all-or-nothing bet).

Step 1:
Make a stemplot.
Is there any sign of a major deviation from Normality?
Which of the following best describes the distribution?

Slightly skewed to the right.

Strongly skewed to the right.

Fairly symmetrical with a dominant single peak.

Two clear symmetrical peaks.

________________________________________

Step 2:
All 20 subjects gambled a fixed amount more often when faced with a sure loss than when faced with a sure win.
Give a 95% confidence interval for the mean percent increase in gambling a fixed amount when faced with a sure loss.

13.7254 to 23.5946

13.9721 to 23.3479

14.6868 to 22.6332

13.8504 to 23.4696

6. (20.09) The one-sample t statistic from a sample of n = 5 observations for the two-sided test of
H0: ? = 50
Ha: ? ? 50
has the value t = 2.50.

Step 1:
What are the degrees of freedom for t?

7

6

4

11

________________________________________

Step 2:
Is the value t = 2.50 statistically significant at the 10% level? At the 5% level?

The value t = 2.50 is significant at the 10% level, but not at the 5% level.

The value t = 2.50 is not significant neither at the 10% level nor at the 5% level.

The value t = 2.50 is significant both at the 10% level and at the 5% level.

The value t = 2.50 is significant at the 5% level, but not at the 10% level.

7. (20.17) We prefer the t procedures to the z procedures for inference about a population mean because

z requires that you know the population standard deviation s.

z can be used only for large samples.

z requires that you can regard your data as an SRS from the population.

8. (20.18) You are testing H0: ? = 100 against Ha: ? < 100 based on an SRS of 9 observations from a Normal population. The data give x = 98 and s = 3. The value of the t statistic is -2. -6. -98. 9. (20.19) You are testing H0: ? = 100 against Ha: ? > 100 based on an SRS of 16 observations from a Normal population. The t statistic is t = 2.13. The degrees of freedom for this statistic are

17.

15.

16.

10. (20.20) You are testing H0: ? = 100 against Ha: ? > 100 based on an SRS of 20 observations from a Normal population. The t statistic is t = 2.5. The P-value for the statistic

falls between 0.05 and 0.10.

falls between 0.01 and 0.05.

is less than 0.01.

11. (20.22) You are testing H0: ? = 0 against Ha: ? ? 0 based on an SRS of 6 observations from a Normal population. What values of the t statistic are statistically significant at the a = 0.001 level?

t > 6.869

t < -5.893 or t > 5.893.

t < -6.869 or t > 6.869.

12. (20.23) Twenty-five adult citizens of the U.S. were asked to estimate the average income of all U.S. households. The mean estimate was = $45,000 and s = $15,000. (Note: the actual average household income at the time of the study was about $68,000.) Assume the 25 adults in the study can be considered an SRS from the population of all adult citizens of the U.S. A 95% confidence interval for the mean estimate of the average income of all U.S. households is

$39,120 to $50,880.

$38,808 to $51,192.

$39,867 to $50,133.

13. (20.24) Twenty-five adult citizens of the U.S. were asked to estimate the average income of all U.S. households. The mean estimate was = $45,000 and s = $15,000. (Note: the actual average household income at the time of the study was about $68,000.) Assume the 25 adults in the study can be considered an SRS from the population of all adult citizens of the U.S. Which of the following would cause the most worry about the validity of a 95% confidence interval that you calculate using this information?

You do not know the population standard deviation s.

You notice that there is a clear outlier in the data.

A stemplot of the data shows a mild right-skew.

14. (20.25) Which of these settings does not allow use of a matched pairs t procedure?

You interview 100 female students in their freshman year and again in their senior year and ask each about the average number of minutes each day she spends using social media.

You interview both spouses in 400 married couples and ask each about the average number of minutes each day they spend using social media.

You interview a sample of 225 unmarried male students and another sample of 225 unmarried female students and ask each about the average number of minutes each day they spend using social media.

15. (20.26) Because the t procedures are robust, the most important condition for their safe use is that

the population standard deviation s is known.

the data can be regarded as an SRS from the population.

the population distribution is exactly Normal.

16. (20.28) In Example 16.1 we developed a 95% z confidence interval for the mean body mass index (BMI) of women aged 20 to 29 years, based on a national random sample of 654 such women.
We assumed there that the population standard deviation was known to be s = 7.5.
In fact, the sample data had mean BMI x = 26.8 and standard deviation s = 7.42.
What is the 95% t confidence interval for the mean BMI of all young women?

26.2303 to 27.3697

26.6036 to 26.9964

26.3184 to 27.2816

26.5313 to 27.0687

17. (20.29) The Trial Urban District Assessment (TUDA) is a government-sponsored study of student achievement in large urban school districts. TUDA gives a reading test scored from 0 to 500. A score of 243 is a ?basic? reading level and a score of 281 is ?proficient.? Scores for a random sample of 1400 eighth-graders in Dallas with standard error 1.0.

Step 1:
We don’t have the 1400 individual scores, but use of the t procedures is surely safe. Why?

Because use of the t procedures is always safe.

Because the data has a strong skew.

Because the sample doesn’t have outliers.

Because the sample size is very large.

Step 2:
Give a 99% confidence interval for the mean score of all Dallas eighth-graders. (Be careful: the report gives the standard error of x, not the standard deviation s.)

238.5 to 241.5

245.4 to 250.6

239.9 to 240.0

237.4 to 242.6

Step 3:
Urban children often perform below the basic level. Is there good evidence that the mean for all Dallas eighth-graders is greater than the basic level?

Yes, there is. The basic level of 243 is not included in the 99% confidence interval.

No, there isn’t. The sample mean of 240 is included in the 99% confidence interval.

No, there isn’t. The basic level of 240 is not included in the 99% confidence interval.

Yes, there is. The sample mean of 243 is included in the 99% confidence interval.

18.
(18.29) The placebo effect is particularly strong in patients with Parkinson’s disease. To understand the workings of the placebo effect, scientists measure activity at a key point in the brain when patients receive a placebo that they think is an active drug and also when no treatment is given. The same six patients are measured both with and without the placebo, at different times

Step 1:
Explain why the proper procedure to compare the mean response to placebo with control (no treatment) is a matched pairs t test.

The proper procedure is a matched pairs t test because the placebo effect is particularly strong in patients with Parkinson’s disease.

The proper procedure is a matched pairs t test because the same six patients are measured both with and without the placebo at different times.

The proper procedure is a matched pairs t test because there are exactly two groups – placebo and no treatment.

The proper procedure is a matched pairs t test because scientists measure activity at a key point in the brain.

________________________________________

Step 2:
The six differences (treatment minus control) had x = -0.326 and s = 0.181. Is there significant evidence of a difference between treatment and control?

Yes, there is (0.01 < P < 0.025).
Yes, there is (0.025 < P < 0.05).
No, there isn’t (P > 0.05).

Yes, there is (P < 0.01).
19.
(20.35) Our bodies have a natural electrical field that is known to help wounds heal.
Does changing the field strength slow healing?
A series of experiments with newts investigated this question.
In one experiment, the two hind limbs of 12 newts were assigned at random to either experimental or control groups.
This is a matched pairs design.
The electrical field in the experimental limbs was reduced to zero by applying a voltage.
The control limbs were left alone.
Here are the rates at which new cells closed a razor cut in each limb, in micrometers per hour:

Note that a higher healing rate means “heals faster” . If the control rate- experimental rate>0, the electrical field on the experimental limb slowed healing compared to the natural (control) electrical field.
Data Set

Step 1:
Make a stemplot of the differences between limbs of the same newt (control limb minus experimental limb).
There is a high outlier.
Choose the correct stemplot.

Stemplot II
-1 3
-0 6
-0 0
0 12
0 5789
1 012
1 0
2 0
2 0
3 1

Stemplot III
-3 1
-2
-2
-1
-1 012
-0 5789
-0 12
0
0 6
14 3

Stemplot I
-1 3
-0 6
-0
0 12
0 5789
1 012
1
2
2
3 1

________________________________________

Step 2:
If ? is the mean difference (control minus experimental) in healing rates, what are the null and alternative hypotheses?

H 0 : ? = 0 vs. Ha : ? < 0
H 0 : ? > 0 vs. Ha : ? < 0
H 0 : ? = 0 vs. Ha : ? ? 0

H 0 : ? = 0 vs. Ha : ? > 0

________________________________________

Step 3:
Carry out a t-test to see if the mean healing rate is significantly lower in the experimental limbs, including all 12 newts.
What are the test statistics and their P-values?

t = 2.08, 0.05 < P < 0.01
t = 2.08, 0.025 < P < 0.05
t = 1.86, 0.05 < P < 0.10
t = 1.86, 0.025 < P < 0.05
________________________________________

Step 4:
Carry out a t-test that omits the outlier.
What are the test statistic and its P-value?

t = 1.79, 0.025 < P < 0.05
t = 1.87, 0.025 < P < 0.05
t = 1.79, 0.05 < P < 0.1
t = 1.87, 0.05 < P < 0.1
________________________________________

Step 5:
Does the outlier have a strong influence on your conclusion?

Yes, it does. At the 5% level, the null hypothesis is rejected when the data includes the outlier but it is not rejected when the outlier is omitted.

Yes, it does. At the 5% level, the null hypothesis is rejected when the outlier is omitted, but it is not rejected when the data includes the outlier.

No, it doesn’t. At the 5% level, the null hypothesis is not rejected neither when the data includes the outlier, nor when the outlier is omitted.

No, it doesn’t, At the 5% level, the null hypothesis is rejected both when the data includes outliers and when the outlier is omitted.

20. (20.38) Here’s a new idea for treating advanced melanoma, the most serious kind of skin cancer.
Genetically engineer white blood cells to better recognize and destroy cancer cells, then infuse these cells into patients.
The subjects in a small initial study were 11 patients whose melanoma had not responded to existing treatments.
One question was how rapidly the new cells would multiply after infusion, as measured by the doubling time in days.
Here are the doubling times:

Data Set

Step 1:
True or False:
There is no severe evidence of non-Normality so t procedures should be safe.

True

False

________________________________________

Step 2:
Give a 90% confidence interval for the mean doubling time.

0.867 to 1.4784

0.9442 to 1.4012

0.9211 to 1.4243

0.9311 to 1.4129

________________________________________

Step 3:
Are you willing to use this interval to make an inference about the mean doubling time in a population of similar patients?

Yes, provided the sample is an SRS, because there are no severe deviations from Normality.

Yes, because the sample size is big enough.

No, because the sample is too small.

No, because the distribution is not Normal.

21. (20.51) The design of controls and instruments affects how easily people can use them. A student project investigated this effect by asking 25 right-handed students to turn a knob (with their right hands) that moved an indicator by screw action. There were two identical instruments, one with a right-hand thread (the knob turns clockwise) and the other with a left-hand thread (the knob turns counterclockwise). Table 18.5 gives the times in seconds each subject took to move the indicator a fixed distance.

Step 1:
Each of the 25 students used both instruments.
True or False: A good way to arrange the experiment is asking each student to use the instrument with the right-hand thread, and then the one with the left-hand thread.

True

False

________________________________________

Step 2:
The project hoped to show that right-handed people find right-hand threads easier to use. Let ? be the mean difference in the time needed to do the action (right minus left). State the hypotheses you should use to reach a conclusion.

H 0: ? = 0
Ha: ? ? 0

H 0: ? = 0
Ha: ? > 0

H 0: ? = 0
Ha: ? < 0
H 0: ? ? 0
Ha: ? = 0

________________________________________

Step 3:
What is your conclusion?

Right-handed people don’t find right-hand threads easier to use (0.05 < P < 0.1).
Right-handed people find right-hand threads easier to use (P < 0.01).
Right-handed people find right-hand threads easier to use (0.01 < P < 0.05).
Right-handed people don’t find right-hand threads easier to use (P > 0.1).

22. (20.44) Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. Here are the sweetness losses ( sweetness before storage minus sweetness after storage) found by 10 tasters for one new cola recipe:
2.1 0.4 0.7 1.9 -0.4
2.4 -1.4 1.4 1 2.5
Take the data from these 10 carefully trained tasters as an SRS from a large population of all trained tasters.
Is there evidence at the 10% level that the cola lost sweetness? If the cola has not lost sweetness, the ratings after should be the same as before it was stored.

The test statisic is t = (??0.001)

No

Yes

 

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