Homework answers / question archive / 1) Which of the following statements are true concerning the mean of the differences between two dependent samples (matched pairs)? Select all that apply

1) Which of the following statements are true concerning the mean of the differences between two dependent samples (matched pairs)? Select all that apply

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1) Which of the following statements are true concerning the mean of the differences between two dependent samples (matched pairs)?

Select all that apply.

A. The methods used to evaluate the mean of the differences between two dependent variables apply if one has 96 heights of men from New York and 96 heights of men from California.

B. If one has twenty matched pairs of sample data, there is a loose requirement that the twenty differences appear to be from a normally distributed population.

C. If one has more than 10 matched pairs of sample data, one can consider the sample to be large and there is no need to check for normality.

D. The requirement of a simple random sample is satisfied if we have dependent pairs of voluntary response data.

E. If one wants to use a confidence interval to test the claim that 14 > 0 with a 0.10 significance level, the confidence interval should have a confidence level of 80%.

2. Assume that the paired data came from a population that is normally distributed. Using a 0.05 significance level and d = x - y, find d, s_d, the t test statistic, and the critical values to test the claim that m_d = 0.

x	7	14	11	7	9	14	10	5
y	10	15	11	12	10	14	5	7

 

d = ______________ (Round to three decimal places as needed.)

S_d =______________ (Round to three decimal places as needed.)

T = ______________ (Round to three decimal places as needed. )

t_a/2 = ±____________ (Round to three decimal places as needed. )

3. Listed below are ages of actresses and actors at the time that they won an award for the categories of Best Actress and Best Actor. Use the sample data to test for a difference between the ages of actresses and actors when they win the award. Use a 0.01 significance level. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal.

Actress's age               19      30      30      49    30

Actor's age                   48      44      62     55     42

In this example, m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the actress's age minus the actor's age. What are the null and alternative hypotheses for the hypothesis test?

A. H₀: m_d = 0

     H₁: m_d < 0

B. H₀: m_d ¹ 0

   H₁: m_d = 0

C. H₀: m_d ¹ 0

    H₁: m_d > 0

D. H₀: m_d = 0

    H₁: m_d ¹ 0

Identify the test statistic.

t=___________ (Round to two decimal places as needed. )

Identify the P-value.

P-value =____________ (Round to three decimal places as needed. )

What is the conclusion based on the hypothesis test?

Since the P-value is (1) _____ than the significance level, (2) ________ the null hypothesis. There

(3)______ sufficient evidence to support the claim that there is a difference between the ages of actresses and actors when they win the award.

1) a) less    b) greater

2) a) fail to reject   b) reject

3) a) is    b) is not

4. Several students were tested for reaction times (in thousandths of a second) using their right and left hands. (Each value is the elapsed time between the release of a strip of paper and the instant that it is caught by the subject.) Results from five of the students are included in the graph to the right. Use a 0.01 significance level to test the claim that there is no difference between the reaction times of the right and left hands.

¹Click the icon to view the reaction time data table.

What are the hypotheses for this test?

Let m_d be the (1) _____ of the right and left hand reaction times.

H₀: m_d (2) _____ 0

H₁: m_d (3) ______ 0

What is the test statistic?

t= _______ (Round to three decimal places as needed. )

Identify the critical value(s). Select the correct choice below and fill the answer box within your choice. (Round to three decimal places as needed. )

A. The critical value is t= ____  .

B. The critical values are t= ± _____ .

What is the conclusion?

There (4) ____ enough evidence to warrant rejection of the claim that there is (5) ______ between the reaction times of the right and left hands.

1: Reaction Time Data Table

Right Hand	Left Hand
116	130
113	141
140	162
187	208
189	203

 

(1) a) mean of the differences 

     b) difference between the means

(2) a) <

     b) £

      c) =

      d) ³

      e) >

       f) ¹

(3) a) £    b) <   c) ³    d) =   e) ¹   f) >

4) a) is not    b) is

(5) a) no difference

     b) a difference

5. The data below are yields for two different types of corn seed that were used on adjacent plots of land. Assume that the data are simple random samples and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the difference between type 1 and type 2 yields. What does the confidence interval suggest about farmer Joe's claim that type 1 seed is better than type 2 seed?

Type 1       2179     1963     2106    2454      2120       1905      2151     1501

Type 2       2072     1909     2079    2413      2115       1941      2101      1444

In this example, m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the type 1 seed yield minus the type 2 seed yield.

The 95% confidence interval is ___________ < m_d <_____.

(Round to two decimal places as needed.)

What does the confidence interval suggest about farmer Joe's claim that type 1 seed is better than type 2 seed?

A. Because the confidence interval includes zero, there is sufficient evidence to support farmer Joe's claim.

B. Because the confidence interval includes zero, there is not sufficient evidence to support farmer Joe's claim.

C. Because the confidence interval only includes positive values and does not include zero, there is not sufficient evidence to support farmer Joe's claim.

D. Because the confidence interval only includes positive values and does not include zero, there is sufficient evidence to support farmer Joe's claim.

6. Researchers collected data on the numbers of hospital admissions resulting from motor vehicle crashes, and results are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Use a 0.05 significance level to test the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.

Friday the 6th:      10      5     12      12      3      4

Friday the 13th:    14    11     14      11      5    10

What are the hypotheses for this test?

Let m_d be the (1) _______ in the numbers of hospital admissions resulting from motor vehicle crashes for the population of all pairs of data.

H₀: m_d (2) _____ 0

H₁: m_d (3) _____ 0

Find the value of the test statistic.

t= _______ (Round to three decimal places as needed.)

Identify the critical value(s). Select the correct choice below and fill the answer box within your choice. (Round to three decimal places as needed.)

A. The critical value is t = ____  .

B. The critical values are t = ± _____  .

State the result of the test. Choose the correct answer below.

A. There is sufficient evidence to warrant rejection of the claim of no effect. Hospital admissions do not appear to be affected.

B. There is not sufficient evidence to warrant rejection of the claim of no effect. Hospital admissions appear to be affected.

C. There is sufficient evidence to warrant rejection of the claim of no effect. Hospital admissions appear to be affected.

D. There is not sufficient evidence to warrant rejection of the claim of no effect. Hospital admissions do not appear to be affected.

(1) a) mean of the differences    b) difference between the means

(2) a) <   b) £  c) =   d) ³   e) >    f) ¹

(3) a) <    b) £  c) =   d) ¹   e) >    f) ³

7. Listed below are systolic blood pressure measurements (mm Hg) taken from the right and left arms of the same woman. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Use a 0.10 significance level to test for a difference between the measurements from the two arms. What can be concluded?

Right arm       144       133      141      138      136

Left arm         176       179      178       152     150

In this example, m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the measurement from the right arm minus the measurement from the left arm. What are the null and alternative hypotheses for the hypothesis test?

A. H₀: m_d ¹ 0

     H₁: m_d > 0

B. H₀: m_d ¹ 0

    H₁: m_d = 0

C. H₀: m_d = 0

    H₁: m_d > 0

D. H₀: m_d = 0

     H₁: m_d ¹ 0

Identify the test statistic.

t= ______ (Round to two decimal places as needed. )

Identify the P-value.

P-value = ______ (Round to three decimal places as needed. )

What is the conclusion based on the hypothesis test?

Since the P-value is (1) ____ than the significance level, (2) _______ the null hypothesis. There

(3) ________ sufficient evidence to support the claim of a difference in measurements between the two arms.

(1) a) less  b) greater

(2) a) fail to reject   b) reject

(3) a) is    b) is not

8. Data on the numbers of hospital admissions resulting from motor vehicle crashes are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the mean of the population of differences between hospital admissions. Use the confidence interval to test the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.

Friday the 6^th             2          4          6        12        10

Friday the 13^th         14        14        10       11         13

In this example, m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the number of hospital admissions on Friday the 6th minus the number of hospital admissions on Friday the 13th. Find the 95% confidence interval.

_______ < m_d < _______

(Round to two decimal places as needed.)

Based on the confidence interval, can one reject the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected?

A. No, because the confidence interval includes zero.

B. No, because the confidence interval does not include zero.

C. Yes, because the confidence interval does not include zero.

D. Yes, because the confidence interval includes zero.

9. When subjects were treated with a drug, their systolic blood pressure readings (in mm Hg) were measured before and after the drug was taken. Results are given in the table below. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Using a 0.05 significance level, is there sufficient evidence to support the claim that the drug is effective in lowering systolic blood pressure?

Before       169      205    195    180    188    164     175    189    183     157    179     168

After         186       143    176    182    179    148     162    155    144     152     147     187

In this example, m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the systolic blood pressure reading before the drug was taken minus the reading after the drug was taken. What are the null and alternative hypotheses for the hypothesis test?

A. H₀: m_d ¹ 0

     H₁: m_d = 0

B. H₀: m_d = 0

    H₁: m_d < 0

C. H₀: m_d = 0

    H₁: m_d > 0

D. H₀: m_d ¹ 0

   H₁: m_d > 0

Identify the test statistic.

t= _________ (Round to two decimal places as needed. )

Identify the P-value.

P-value = _______ (Round to three decimal places as needed. )

Since the P-value is (1) _________ than the significance level, (2) _______ H₀. There is

(3)______ evidence to support the claim that the drug is effective in lowering systolic blood pressure.

(1) a) greater    b) less

2) a) fail to reject    b) reject

3) a) sufficient     b) insufficient

10. A study was conducted to measure the effectiveness of hypnotism in reducing pain. The measurements are centimeters on a pain scale before and after hypnosis. Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval for the mean of the "before — after" differences. Does hypnotism appear to be effective in reducing pain?

Before       11.7        4.2        9.0      12.0       9.8       8.1        4.4       7.0

After            6.8         2.5        7.3        8.4       8.7      6.3         3.6       2.6

Construct a 95% confidence interval for the mean of the "before — after" differences.

______ < m_d < ______ (Round to two decimal places as needed. )

Does hypnotism appear to be effective in reducing pain?

A. Yes, because the confidence interval does not include zero and is entirely greater than zero.

B. No, because the confidence interval includes zero.

C. No, because the confidence interval does not include zero and is entirely greater than zero.

D. Yes, because the confidence interval includes zero.

11. Listed below are the heights of candidates who won elections and the heights of the candidates with the next highest number of votes. The data are in chronological order, so the corresponding heights from the two lists are matched. Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the mean of the population of all "winner/runner-up" differences. Does height appear to be an important factor in winning an election?

Winner          76       71        71       72       78      76      73       67

Runner-Up   73       72        70       70        74       74     69       68

Construct the 95% confidence interval. (Subtract the height of the runner-up from the height of the winner to find the difference, d.)

_______ < m_d < ______ (Round to two decimal places as needed. )

Based on the confidence interval, does height appear to be an important factor in winning an election?

A. Yes, because the confidence interval does not include zero.

B. No, because the confidence interval does not include zero.

C. No, because the confidence interval includes zero.

D. Yes, because the confidence interval includes zero.

12. Refer to the data set in the accompanying table. Assume that the paired sample data is a simple random sample and the differences have a distribution that is approximately normal. Use a significance level of 0.01 to test for a difference between the weights of discarded paper (in pounds) and weights of discarded plastic (in pounds).

²Click the icon to view the data.

In this example, m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the weight of discarded paper minus the weight of discarded plastic for a household. What are the null and alternative hypotheses for the hypothesis test?

A. H₀: m_d ¹ 0

     H₁: m_d = 0

B. H₀: m_d = 0

    H₁: m_d < 0

C. H₀: m_d = 0

    H₁: m_d ¹ 0

D. H₀: m_d ¹ 0

   H₁: m_d > 0

Identify the test statistic.

t= ________ (Round to two decimal places as needed. )

Identify the P-value.

P-value = ________ (Round to three decimal places as needed. )

What is the conclusion based on the hypothesis test?

Since the P-value is (1) _______ than the significance level, (2) ______ the null hypothesis. There

(3) ______ Sufficient evidence to support the claim that there is a difference between the weights of discarded paper and discarded plastic.

2: More Info

 

Household	Paper	Plastic
1	15.09	9.11
2	16.08	14.36
3	17.65	11.26
4	7.72	3.86
5	8.82	11.89
6	6.96	7.60
7	9.19	3.74
8	13.61	8.95
9	11.42	12.81
10	9.41	3.36
11	12.32	11.17
12	5.86	3.91
13	7.57	5.92
14	6.44	8.40
15	9.83	6.26
16	6.05	2.73
17	2.41	1.13
18	9.45	3.02
19	12.43	8.57
20	6.33	3.86
21	11.36	10.25
22	9.55	9.20
23	13.05	12.31
24	14.33	6.43
25	6.16	5.88
26	11.08	12.47
27	16.39	9.70
28	8.72	9.20
29	13.31	19.70
30	6.38	8.82
Household	Paper	Plastic

 

(1) a) greater    b) less

(2)  a) reject     b) fail to reject

(3) a) is not     b) is

13. Refer to the data set in the accompanying table. Assume that the paired sample data is a simple random sample and the differences have a distribution that is approximately normal. Use a significance level of 0.05 to test for a difference between the number of words spoken in a day by each member of 30 different couples.

³Click the icon to view the data.

In this example m_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the number of words spoken by a male minus the number of words spoken by a female in a couple. What are the null and alternative hypotheses for the hypothesis test?

A. H₀: m_d ¹ 0

     H₁: m_d > 0

B. H₀: m_d = 0

    H₁: m_d < 0

C. H₀: m_d = 0

    H₁: m_d ¹ 0

D. H₀: m_d ¹ 0

   H₁: m_d = 0

Identify the test statistic.

t= ______ (Round to two decimal places as needed. )

Identify the P-value.

P-value = (Round to three decimal places as needed. )

What is the conclusion based on the hypothesis test?

Since the P-value is (1) ________ then the significance level, (2) ______ the null hypothesis. There

(3) _____ sufficient evidence to support the claim that there is a difference between the number of words spoken in a day by each member of 30 different couples.

3: More Info

Couple	Male	Female
1	2800	5922
2	13,050	12,306
3	12,430	8568
4	20,120	18,354
5	9450	3024
6	12,320	11,172
7	6670	6090
8	6440	8400
9	11,360	10,248
10	9190	3738
11	3270	630
12	11,420	12,810
13	6330	3864
14	16,080	14,364
15	6380	8820
16	6160	5880
17	8820	11,886
18	12,730	14,826
19	2410	1134
20	6050	2730
21	17,650	11,256
22	16,390	9720
23	9550	9198
24	7980	6090
25	6960	7602
26	7720	3864
27	9830	6258
28	11,080	12,474
29	13,310	19,698
30	15,090	9114
Couple	Male	Female

 

(1) a) greater    b) less

(2)   a) reject    b) fail to reject

(3) a) is     b) is not

14. Which of the following is NOT a principle of making inferences from dependent samples?

Choose the correct answer below.

A. There is some relationship whereby each value in one sample is paired with a corresponding value in the other sample.

B. The hypothesis test and confidence interval are equivalent in the sense that they result in the same conclusion.

C. The t-distribution serves as a reasonably good approximation for inferences from dependent samples.

D. Testing the null hypothesis that the mean difference equals 0 is not equivalent to determining whether the confidence interval includes 0.

15. Which of the following is NOT a requirement of testing a claim about the mean of the differences from dependent samples?

Choose the correct answer below.

A. Either the number of pairs of sample data is larger than 30 or the pairs have differences that are from a population having a distribution that is approximately normal, or both.

B. The samples are simple random samples.

C. The degrees of freedom are n- 2.

D. The sample data are dependent.

16. What design principle is stressed for experiments or observational studies?

Choose the correct answer below.

A. When using paired data, keep the sample very small in order to keep the data manageable.

B. Using dependent samples with paired data is generally better than using two independent samples.

C. The methods for dependent samples can be used for any matched pairs.

D. Using two independent samples is the best choice for accurate results.