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.
