Mastery 7

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Determine the critical values for the following tests of the population mean with an unknown population standard deviation. The analysis is based on 26 observations drawn from a normally distributed population at a 1% level of significance. Use Table 2. (Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.)

Critical Value

H0: μ ≤ 46 versus HA: μ > 46

b. H0: μ = 13.8 versus HA: μ ≠ 13.8 ±

c. H0: μ ≥ 4.2 versus HA: μ < 4.2

d. H0: μ = 17 versus HA: μ ≠ 17 ±

rev: 08_21_2013_QC_33738

The margarita is one of the most common tequila-based cocktails, made with tequila, triple sec, and lime juice, often served with salt on the glass rim. A manager at a local bar is concerned that the bartender is not using the correct amounts of the three ingredients in more than 50% of margaritas. He secretly observed the bartender and found that he used the CORRECT amounts in only 20 out of the 44 margaritas in the sample. Use the critical value approach to test if the manager's suspicion is justified at α = 0.01. Let p represent the proportion of all margaritas made by the bartender that have INCORRECT amounts of the three ingredients. Use Table 1.

Select the null and the alternative hypotheses.

H0: p = 0.50; HA: p ≠ 0.50

H0: p ≥ 0.50; HA: p < 0.50

H0: p ≤ 0.50; HA: p > 0.50

Calculate the sample proportion. (Round your answer to 3 decimal places.)

Sample proportion

Calculate the value of test statistic. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)

Test statistic

Calculate the critical value. (Round your answer to 2 decimal places.)

Critical value

What is the conclusion?

The manager's suspicion is not justified since the value of the test statistic does not fall in the rejection region.

The manager’s suspicion is not justified since the value of the test statistic falls in the rejection region.

The manager's suspicion is justified since the value of the test statistic falls in the rejection region.

The manager’s suspicion is justified since the value of the test statistic does not fall in the rejection region.

rev: 09_20_2012, 08_02_2013_QC_32363, 08_23_2013_QC_33738, 11_22_2013_QC_41011

A politician claims that he is supported by a clear majority of voters. In a recent survey, 29 out of 49 randomly selected voters indicated that they would vote for the politician. Use a 10% significance level for the test. Use Table 1.

Select the null and the alternative hypotheses.

H0: p = 0.50; HA: p ≠ 0.50

H0: p ≤ 0.50; HA: p > 0.50

H0: p ≥ 0.50; HA: p < 0.50

Calculate the sample proportion. (Round your answer to 3 decimal places.)

Sample proportion

Calculate the value of test statistic. (Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

Calculate the p-value of the test statistic. (Round intermediate calculations to 4 decimal places. Round "z" value to 2 decimal places and final answer to 4 decimal places.)

p-value

What is the conclusion?

Do not reject H0; the politician is not supported by a clear majority

Do not reject H0; the politician is supported by a clear majority

Reject H0; the politician is not supported by a clear majority

Reject H0; the politician is supported by a clear majority

rev: 08_02_2013_QC_32363, 08_23_2013_QC_33738

Consider the following hypotheses:

H0: μ ≥ 134

HA: μ < 134

A sample of 62 observations results in a sample mean of 132. The population standard deviation is known to be 32. Use Table 1.

What is the critical value for the test with α = 0.10 and with α = 0.01? (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

Critical Value

α = 0.10

α = 0.01

b-1.

Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

b-2.

Does the above sample evidence enable us to reject the null hypothesis at α = 0.10?

No since the value of the test statistic is not less than the negative critical value.

Yes since the value of the test statistic is less than the negative critical value.

Yes since the value of the test statistic is not less than the negative critical value.

No since the value of the test statistic is less than the negative critical value.

Does the above sample evidence enable us to reject the null hypothesis at α = 0.01?

No since the value of the test statistic is not less than the negative critical value.

Yes since the value of the test statistic is less than the negative critical value.

Yes since the value of the test statistic is not less than the negative critical value.

No since the value of the test statistic is less than the negative critical value.

rev: 08_02_2013_QC_32363, 08_23_2013_QC_33738, 11_10_2014_QC_58468

Consider the following hypotheses:

H0: μ = 340

HA: μ ≠ 340

The population is normally distributed with a population standard deviation of 63. Use Table 1.

Use a 10% level of significance to determine the critical value(s) of the test. (Round your answer to 2 decimal places.)

Critical value(s) ±

b-1.

Calculate the value of the test statistic with formula287.mml = 371 and n = 50. (Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

b-2. What is the conclusion at α = 0.10?

Do not reject H0 since the value of the test statistic is smaller than the critical value.

Do not reject H0 since the value of the test statistic is greater than the critical value.

Reject H0 since the value of the test statistic is smaller than the critical value.

Reject H0 since the value of the test statistic is greater than the critical value.

Use a 5% level of significance to determine the critical value(s) of the test. (Round your answer to 2 decimal places.)

Critical value(s) ±

d-1.

Calculate the value of the test statistic with formula287.mml = 318 and n = 50. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

d-2. What is the conclusion at α = 0.05?

Reject H0 since the value of the test statistic is not less than the negative critical value.

Reject H0 since the value of the test statistic is less than the negative critical value.

Do not reject H0 since the value of the test statistic is not less than the negative critical value.

Do not reject H0 since the value of the test statistic is less than the negative critical value.

rev: 08_21_2013_QC_33738

A local bottler in Hawaii wishes to ensure that an average of 12 ounces of passion fruit juice is used to fill each bottle. In order to analyze the accuracy of the bottling process, he takes a random sample of 76 bottles. The mean weight of the passion fruit juice in the sample is 11.83 ounces. Assume that the population standard deviation is 1.22 ounce. Use Table 1.

Use the critical value approach to test the bottler's concern at α = 0.10.

Select the null and the alternative hypotheses for the test.

H0: μ ≥ 12; HA: μ < 12

H0: μ = 12; HA: μ ≠ 12

H0: μ ≤ 12; HA: μ > 12

b-1.

Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

b-2.

Find the critical value(s). (Round your answer to 2 decimal places.)

Critical value(s) ±

b-3.

What is the conclusion?

Do not reject H0 since the value of the test statistic is less than the negative critical value.

Do not reject H0 since the value of the test statistic is not less than the negative critical value.

Reject H0 since the value of the test statistic is less than the negative critical value.

Reject H0 since the value of the test statistic is not less than the negative critical value.

Make a recommendation to the bottler.

The accuracy of the bottling process is

rev: 08_02_2013_QC_32363, 08_23_2013_QC_33738

Consider the following hypotheses:

H0: μ ≥ 80

HA: μ < 80

The population is normally distributed. A sample produces the following observations:

72 63 62 80 62 75

Use the critical value approach to conduct the test at a 1% level of significance. Use Table 2.

Find the mean and the standard deviation. (Round intermediate calculations to 4 decimal places. Round your answers to 2 decimal places.)

Mean

Standard deviation

Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

Calculate the critical value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 3 decimal places.)

Critical value

d. What is the conclusion?

Do not reject H0 since the value of the test statistic is less than the negative critical value.

Do not reject H0 since the value of the test statistic is not less than the negative critical value.

Reject H0 since the value of the test statistic is less than the negative critical value.

Reject H0 since the value of the test statistic is not less than the negative critical value.

rev: 08_02_2013_QC_32363, 08_23_2013_QC_33738, 11_22_2013_QC_41028

Consider the following hypotheses:

H0: p ≥ 0.34

HA: p < 0.34

Which of the following sample information enables us to reject the null hypothesis at α = 0.01 and at

α = 0.10? Use Table 1.

α = 0.01 α = 0.10

a. x = 32; n = 100

b. x = 64; n = 280

c. formula225.mml = 0.31; n = 42

d. formula226.mml = 0.31; n = 451

rev: 08_02_2013_QC_32363

A polygraph (lie detector) is an instrument used to determine if the individual is telling the truth. These tests are considered to be 98% reliable. In other words, if an individual lies, there is a 0.98 probability that the test will detect a lie. Let there also be a 0.015 probability that the test erroneously detects a lie even when the individual is actually telling the truth. Consider the null hypothesis, "the individual is telling the truth," to answer the following questions.

a. What is the probability of Type I error? (Round your answer to 3 decimal places.)

Probability

b. What is the probability of Type II error? (Round your answer to 2 decimal places.)

Probability

10.

Consider the following hypotheses:

H0: p ≥ 0.47

HA: p < 0.47

Compute the p-value based on the following sample information. Use Table 1. (Round intermediate calculations to 4 decimal places. Round "z" value to 2 decimal places and final answers to 4 decimal places.)

p-value

x = 46; n = 118

b. x = 112; n = 311

c. formula225.mml = 0.38; n = 53

d. formula226.mml = 0.38; n = 430

rev: 08_21_2013_QC_33738

11.

Consider the following hypotheses:

H0: μ = 14

HA: μ ≠ 14

The population is normally distributed. A sample produces the following observations:

9 17 19 11 15 13 16

Use the p-value approach to conduct the test at a 1% level of significance. Use Table 2.

PictureClick here for the Excel Data File

Observations
9
17
19
11
15
13
16

Find the mean and the standard deviation. (Round intermediate calculations to 4 decimal places. Round your answers to 2 decimal places.)

Mean

Standard deviation

Calculate the value of the test statistic. (Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

Approximate the p-value of the test statistic.

0.05 < p-value < 0.10

0.10 < p-value < 0.20

p-value > 0.20

d. What is the conclusion?

Reject H0 since the p-value is [a(30)] than α.

Reject H0 since the p-value is [a(31)] than α.

Do not reject H0 since the p-value is [a(30)] than α.

Do not reject H0 since the p-value is [a(31)] than α.

rev: 08_21_2013_QC_33738

12.

A retailer is looking to evaluate its customer service. Management has determined that if the retailer wants to stay competitive, then it will have to have at least a 86% satisfaction rate among its customers. Management will take corrective actions if the satisfaction rate falls below 86%. A survey of 1,400 customers showed that 1,176 were satisfied with their customer service. Use Table 1.

Select the hypotheses to test if the retailer needs to improve its services.

H0: p ≤ 0.86; HA: p > 0.86

H0: p = 0.86; HA: p ≠ 0.86

H0: p ≥ 0.86; HA: p < 0.86

What is the value of the appropriate test statistic? (Negative value should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

Compute the p-value. (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

p-value

d. What is the conclusion?

The management will not take corrective action.

The management will take corrective action.

rev: 08_02_2013_QC_32363, 08_23_2013_QC_33738

13.

Access the hourly wage data on the below Excel Data File (Hourly Wage). An economist wants to test if the average hourly wage is less than $29.

PictureClick here for the Excel Data File

Hourly Wage	EDUC	EXPER	AGE	Gender
24.42	11	2	40	1
10.38	4	1	39	0
35.63	4	2	38	0
37.10	5	9	53	1
21.72	6	15	59	1
37.59	6	12	36	1
27.90	9	5	45	0
16.53	4	12	37	0
28.44	5	14	37	1
24.22	11	3	43	1
39.09	8	5	32	0
30.34	9	18	40	1
28.40	7	1	49	1
22.15	4	10	43	0
28.86	1	9	31	0
34.02	9	22	45	0
40.30	11	3	31	1
32.02	4	14	55	0
14.05	6	5	30	1
13.97	9	3	28	0
16.44	6	15	60	1
30.30	4	13	32	0
33.18	4	9	58	1
22.65	5	4	28	0
34.86	6	5	40	1
23.64	6	2	37	0
23.59	4	18	52	1
12.17	6	4	44	0
30.13	6	4	57	0
31.36	9	3	30	1
13.80	5	8	43	0
39.85	7	6	31	1
10.31	4	3	33	0
22.67	6	23	51	1
13.47	4	15	37	0
28.74	4	9	45	0
34.20	6	3	55	0
13.07	5	14	57	0
39.01	9	16	36	1
39.47	4	20	60	1
37.99	4	5	35	0
35.33	9	10	34	0
40.62	5	4	28	1
28.94	6	1	25	0
12.13	7	10	43	1
34.49	9	2	42	1
14.23	4	17	47	0
43.96	11	2	46	1
34.70	4	15	52	0
15.48	8	11	64	0

Select the null and the alternative hypotheses for the test.

H0: μ ≥ 29; HA: μ < 29

H0: μ = 29; HA: μ ≠ 29

H0: μ ≤ 29; HA: μ > 29

Use the Excel function Z.TEST to calculate the p-value. Assume that the population standard deviation is $6. (Round your answer to 4 decimal places.)

p-value

At α = 0.10 what is the conclusion?

Reject H0; the hourly wage is less than $29.

Reject H0; the hourly wage is not less than $29.

Do not reject H0; the hourly wage is less than $29.

Do not reject H0; the hourly wage is not less than $29.

rev: 08_02_2013_QC_32363, 08_23_2013_QC_33738

14.

Consider the following hypotheses:

H0: μ = 5,900

HA: μ ≠ 5,900

The population is normally distributed with a population standard deviation of 620. Compute the value of the test statistic and the resulting p-value for each of the following sample results. For each sample, determine if you can "reject/do not reject" the null hypothesis at the 10% significance level. Use Table 1. (Negative values should be indicated by a minus sign. Round intermediate calculations to 4 decimal places. Round "test statistic" values to 2 decimal places and "p-value" to 4 decimal places.)

Test Statistic p-value

a. formula279.mml = 5,980; n = 125

b. formula279.mml = 5,980; n = 305

c. formula279.mml = 5,630; n = 34

d. formula279.mml = 5,690; n = 34

rev: 08_21_2013_QC_33738

15.

You would like to determine if more than 55% of the observations in a population are below 10. At α = 0.01, conduct the test on the basis of the following 20 sample observations: Use Table 1.

PictureClick here for the Excel Data File

Observations
8	6	7	8	14	5	14	13	12	6
6	9	10	11	9	7	6	6	10	12

Select the null and the alternative hypotheses.

H0: p ≤ 0.55; HA: p > 0.55

H0: p = 0.55; HA: p ≠ 0.55

H0: p ≥ 0.55; HA: p < 0.55

Calculate the sample proportion. (Round your answer to 2 decimal places.)

Sample proportion

Calculate the value of test statistic. (Round intermediate calculations to 4 decimal places. Round your answer to 2 decimal places.)

Test statistic

Calculate the p-value of the test statistic. (Round intermediate calculations to 4 decimal places. Round "z" value to 2 decimal places and final answer to 4 decimal places.)

p-value

What is the conclusion?

Reject H0 since the p-value is smaller than α.

Reject H0 since the p-value is greater than α.

Do not reject H0 since the p-value is smaller than α.

Do not reject H0 since the p-value is greater than α.