Stats 7.2 - 7.3

In hypothesis​ testing, does choosing between the critical value method or the​ P-value method affect your​ conclusion? Explain.

No, because both involve comparing the test​ statistic's probability with the level of significance.

Find the critical​ value(s) and rejection​ region(s) for the type of​ z-test with level of significance α. Right​-tailed ​test, α = 0.05

1- α The critical​ value(s) is/are z = 1.65. The rejection region is z > 1.65.

Find the critical​ value(s) and rejection​ region(s) for the indicated​ t-test, level of significance α​, and sample size n. Right​-tailed ​test, α = 0.01​, n = 28

2.473 t > 2.473

Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. If a hypothesis test is​ performed, how should you interpret a decision that​ (a) rejects the null hypothesis or​ (b) fails to reject the null​ hypothesis? A government agency claims that at most 84​% of​ full-time workers earn less than ​$535 per week.

Since the claim contains a statement of​ equality, it represents the null hypothesis. There is enough evidence to reject the claim that at most 84​% of​ full-time workers earn less than ​$535 per week. There is not enough evidence to reject the claim that at most 84​% of​ full-time workers earn less than ​$535 per week.

Critical Value

Standard Normal Table ½α Left tail: α Right tail: 1 - α Two tails: ½α, 1 - ½α

Standardized test statistic formula (STS)

z = xbar - µ/σ/√n

Find the critical​ value(s) and rejection​ region(s) for the indicated​ t-test, level of significance α​, and sample size n. Two​-tailed ​test, α = 0.10​, n = 22

-1.721, 1.721 t < -1.721 and t > 1.721

Find the critical​ value(s) and rejection​ region(s) for the indicated​ t-test, level of significance α​, and sample size n. Left​-tailed ​test, α = 0.01​, n = 6

-3.365 t < -3.365

Test the claim about the population​ mean, µ​, at the given level of significance using the given sample statistics. ​Claim: µ = 40​; α = 0.02​; σ = 3.16. Sample​ statistics: x bar = 38.6​, n = 53 Identify the null and alternative hypotheses. Calculate the standardized test statistic. Determine the critical​ value(s). Determine the outcome and conclusion of the test.

H₀: µ = 40 Ha​: µ ≠ 40 STS = -3.23 The critical values are ± 2.33. Reject H₀. At the 2​% significance​ level, there is enough evidence to reject the claim.

Test the claim about the population​ mean, µ​, at the given level of significance using the given sample statistics. ​Claim: µ = 30​; α = 0.01​; σ = 3.78. Sample​ statistics: xbar = 29.2​, n = 57 Identify the null and alternative hypotheses. Calculate the standardized test statistic. Determine the critical​ value(s). Determine the outcome and conclusion of the test.

H₀​: µ = 30 Ha​: µ ≠ 30 z = 29.2 - 30/3.78/√57 z = -0.8/.500673 z = -1.60 The critical values are ± 2.58. Fail to reject Upper H₀. At the 1​% significance​ level, there is not enough evidence to reject the claim.

An environmentalist estimates that the mean waste recycled by adults in the country is more than 1 pound per person per day. You want to test this claim. You find that the mean waste recycled per person per day for a random sample of 15 adults in the country is 1.4 pounds and the standard deviation is 0.4 pound. At α = 0.10​, can you support the​ claim? Assume the population is normally distributed. Write the claim mathematically and identify H₀ and Ha. Find the critical​ value(s) and identify the rejection​ region(s). Find the standardized test statistic. Decide whether to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim.

H₀​: µ ≤ 1 Ha​: µ> 1 t₀ =1.345(t- distribution) t = 3.87 Reject H₀ because the standardized test statistic is in the rejection region. There is sufficient evidence to support the claim that the mean waste recycled is more than 1 pound per person per day.

Use technology to help you test the claim about the population​ mean, mu​, at the given level of​ significance, alpha​, using the given sample statistics. Assume the population is normally distributed. ​Claim: µ > 1190​; α = 0.07​; σ = 195.97. Sample​ statistics: xbar = 1216.05​, n = 250. Identify the null and alternative hypotheses. Calculate the standardized test statistic. Determine the​ P-value. Determine the outcome and conclusion of the test.

H₀​: µ ≤ 1190 Ha​: µ > 1190 z = 1216.05 - 1190/195.97/√250 z = 26.05/12.3942 z = 2.10 P = .018 Reject H₀. At the 7​% significance​ level, there is enough evidence to support the claim.

Use technology and a​ t-test to test the claim about the population mean mu at the given level of significance alpha using the given sample statistics. Assume the population is normally distributed. Claim: µ ≥ 72​; α = 0.05 Sample​ statistics: xbar = 74.7​, s = 3.5​, n = 27 What are the null and alternative​ hypotheses? What is the value of the standardized test​ statistic? What is the​ P-value of the test​ statistic? Decide whether to reject or fail to reject the null hypothesis.

H₀​: µ ≤ 72 HA​: µ > 72 The standardized test statistic = 4.01 P-value = 0 ​ Reject H₀. There is enough evidence to support the claim.

Use technology and a​ t-test to test the claim about the population mean µ at the given level of significance α using the given sample statistics. Assume the population is normally distributed. Claim: µ > 79​; α = 0.05 Sample​ statistics: xbar = 79.4​, s = 3.3​, n = 23. What are the null and alternative​ hypotheses? What is the value of the standardized test​ statistic?

H₀​: µ ≤ 79 HA​: µ > 79 The standardized test statistic = .58 ​P-value = .281 Fail to reject H₀. There is not enough evidence to support the claim.

Use a​ t-test to test the claim about the population mean mu at the given level of significance alpha using the given sample statistics. Assume the population is normally distributed. ​Claim: µ ≠ 29​; α = 0.10 Sample​ statistics: xbar = 30.1​, s = 4.4​, n = 11 What are the null and alternative​ hypotheses? Choose the correct answer below. What is the value of the standardized test​ statistic? What is the​ P-value of the test​ statistic?

H₀​​: µ = 29 Ha​: µ ≠ 29 The standardized test statistic = .83 ​P-value = .426 (t-distribution) Fail to reject H₀. There is not enough evidence to support the claim.

Interpret the decision in the context of the original claim.

If the claim is the null hypothesis and H₀ is​ rejected, then there is enough evidence to reject the claim. If H₀ is not​ rejected, then there is not enough evidence to reject the claim. If the claim is the alternative hypothesis and H₀ is​ rejected, then there is enough evidence to support the claim. If H₀ is not​ rejected, then there is not enough evidence to support the claim.

Explain the difference between the​ z-test for µ using rejection​ region(s) and the​ z-test for µ using a​ P-value.

In the ​z-test using rejection​ region(s), the test statistic is compared with critical values. The​ z-test using a​ P-value compares the​ P-value with the level of significance α.

When P > α​, does the standardized test statistic lie inside or outside of the rejection​ region(s)? Explain your reasoning.

Outside; When the standardized test statistic is inside the rejection​ region, P < α.

Find the​ P-value for the indicated hypothesis test with the given standardized test​ statistic, z. Decide whether to reject H₀ for the given level of significance α. ​Right-tailed test with test statistic z = 1.65 and α = 0.07

P-value = .0495 Reject H₀ If the​ P-value is less than or equal to the level of​ significance, then reject H₀. If the​ P-value is greater than the level of​ significance, then fail to reject H₀.

Find the​ P-value for a​ left-tailed hypothesis test with a test statistic of z = -1.38. Decide whether to reject H₀ if the level of significance is α = 0.05.

P-value = 0.0838 Since P > α​, fail to reject H₀.

Reject or Fail to reject

Reject H₀ if the​ P-value is less than or equal to α. ​Otherwise, fail to reject H₀. The rejection region is the​ blue-shaded portion on the graph. If z is in the rejection​ region, reject H₀. ​Otherwise, fail to reject H₀.

Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. If a hypothesis test is​ performed, how should you interpret a decision that​ (a) rejects the null hypothesis or​ (b) fails to reject the null​ hypothesis? A scientist claims that the mean incubation period for the eggs of a species of bird is at least 40 days.

Since the claim contains a statement of​ equality, it represents the null hypothesis. There is sufficient evidence to reject the claim that the mean incubation period for the eggs of a species of bird is at least 40 days. There is insufficient evidence to reject the claim that the mean incubation period for the eggs of a species of bird is at least 40 days.

Use the calculator displays to the right to make a decision to reject or fail to reject the null hypothesis at a significance level of α = 0.01.

Since the​ P-value is greater than α​, fail to reject the null hypothesis.

P value

Standard Normal Table of the STS.

Find the critical​ value(s) for a​ left-tailed z-test with α = 0.02.

The critical​ value(s) is(are) -2.05.

A car company says that the mean gas mileage for its luxury sedan is at least 20 miles per gallon​ (mpg). You believe the claim is incorrect and find that a random sample of 6 cars has a mean gas mileage of 17 mpg and a standard deviation of 5 mpg. At α = 0.05​, test the​ company's claim. Assume the population is normally distributed. Which sampling distribution should be used and​ why? State the appropriate hypotheses to test. What is the value of the standardized test​ statistic? What is the critical​ value? What is the outcome and the conclusion of this​ test?

Use a ​t-sampling distribution because the population is​ normal, and σ is unknown. H₀​: µ ≥ 20 Ha​: µ < 20 The standardized test statistic = -1.47 The critical value = -2.015 Fail to reject H₀. At the 5​% significance​ level, there is insufficient evidence to reject the car​ company's claim that the mean gas mileage for the luxury sedan is at least 20 miles per gallon.

A null hypothesis is rejected with a level of significance of 0.05. Is it also rejected with a level of significance of​ 0.10? Explain.

Yes, the claim would be rejected because the​ P-value would be less than 0.10.

State whether the standardized test statistic z indicates that you should reject the null hypothesis. Two tailed: Z₀ = -1.960, 1.960 ​(a) z = 1.922 ​(b) z = 2.043 ​(c) z = -1.778 ​(d) z = -2.124

a) Fail to reject H₀ because -1.960 < z < 1.960. b) Reject H₀ because z > 1.960. c) Fail to reject H₀ because -1.960 < z < 1.960. d) Reject H₀ because z < -1.960

State whether the standardized test statistic z indicates that you should reject the null hypothesis. Left tailed: Z₀ = -1.285 ​(a) z = 1.247 ​(b) z = -1.337 ​(c) z = -1.524 ​(d) z = -1.195

a) Fail to reject H₀ because z > -1.285 b) Reject H₀ because z < -1.285 c) Reject H₀ because z < -1.285 d) Fail to reject H₀ because z > -1.285

A random sample of 76 eighth grade​ students' scores on a national mathematics assessment test has a mean score of 293. This test result prompts a state school administrator to declare that the mean score for the​ state's eighth graders on this exam is more than 285. Assume that the population standard deviation is 32. At α = 0.07​, is there enough evidence to support the​ administrator's claim? Complete parts​ (a) through​ (e). a) Write the claim mathematically and identify H₀ and Ha. b) Find the standardized test statistic​ z, and its corresponding area. c) Find the​ P-value. d) Decide whether to reject or fail to reject the null hypothesis. e) Interpret your decision in the context of the original claim.

a) H₀​: µ ≤ 285 Ha​: µ > 285 ​(claim) b) z= 293 - 285/32/√76 z = 8/3.6707 z = 2.18 c) P = 0.015 d) Reject H₀ e) At the 7​% significance​ level, there (is) enough evidence to (support) the​ administrator's claim that the mean score for the​ state's eighth graders on the exam is more than 285.

Determine whether to reject or fail to reject H₀ at the level of significance of​ a) α = 0.04 and​ b) α = 0.02. H₀​: µ = 148​, Ha​: µ ≠148​, and P = 0.0209.​ a) Do you reject or fail to reject H₀ at the 0.04 level of​ significance? b) Do you reject or fail to reject H₀ at the 0.02 level of​ significance?

a) Reject H₀ because P ≤ 0.04 b) Fail to reject H₀ because P > 0.02.

State whether the standardized test statistic z indicates that you should reject the null hypothesis. Right tailed: Z₀ = 1.285 (a) z = 1.311 ​(b) z = 1.065 ​(c) z = -1.148 ​(d) z = 1.525

a) Reject H₀ because z > 1.285. b) Fail to reject H₀ because z < 1.285. c) Fail to reject H₀ because z < 1.285. d) Reject H₀ because z > 1.285.

Find the critical​ value(s) and rejection​ region(s) for the type of​ z-test with level of significance α. Two​-tailed ​test, α = 0.03

½α, 1-½α. The critical​ value(s) is/are z = -2.17, 2.17. The rejection regions are z< -2.17 and z > 2.17.