Chapter 7: Introduction to Hypothesis Testing

*Finding Critical Values in the Standard Normal Distribution* (p. 368)

*1.)* Specify the level of significance *α*. *2.)* Determine whether the test is left-tailed, right-tailed, or two-tailed. *3.)* Find the critical value(s) *"z"₀*. When the hypothesis test is: ~ *[a]:* *left-tailed*, find the *z*-score that corresponds to an area of *α*. ~ *[b]:* *right-tailed*, find the *z*-score that corresponds to an area of 1 - *α*. ~ *[c]:* *two-tailed*, find the *z*-score that corresponds to ½×*α* and 1 - (½×*α*). *4.)* Sketch the standard normal distribution. Draw a vertical line at each critical value and shade the rejection region(s).

*Finding Critical Values in a "t" - Distribution* (p. 377)

*1.)* Specify the level of significance *α*. *2.)* Identify the degrees of freedom: d.f. = *n* - 1. *3.)* Find the critical value(s) using Table 5 in Appendix B in the row *n* - 1 degrees of freedom. When the hypothesis test is: ~ *[a]:* *left-tailed*, use the *One Tail, "α"* column with a negative sign. ~ *[b]:* *right-tailed*, use the *One Tail, "α"* column with a positive sign. ~ *[c]:* *two-tailed*, use the *Two Tail, "α"* column with both a negative and a positive sign.

Steps for Hypothesis Testing (p. 357)

*1.)* State the claim mathematically and verbally. Identify the null and alternative hypotheses. *H₀*: *?* *H*∨*a*: *?* *2.)* Specify the level of significance. *α* = *?* *3.)* Determine the standardized sampling distribution and sketch its graph. (Since I don't have Quizlet+, I can't insert the image of the actual normal curve; ergo, I pasted (some of) the information.) ~~ *The sampling distribution is based on the assumption that "H₀" is true.* *4.)* Calculate the test statistic and its corresponding standardized test statistic. Add it to your sketch. (Since I don't have Quizlet+, I can't insert the image of the actual normal curve; ergo, I pasted (some of) the information.) *Standardized test statistic* = *blue, shaded* area to the *right* of *0*. *5.)* Find the *P*-value. *6.)* Use this decision rule: Is the *P*-value less than or equal to the level of significance? *Yes:* Reject *H₀*. *No:* Fail to reject *H₀*. *7.)* Write a statement to interpret the decision in the context of the original claim.

*Using the "t"-Test for a Mean "μ" ("σ" Unknown)* (p. 379)

*In Words:* *1.)* Verify that *σ* is *not* known, the sample is random, and either the population is normally distributed or *n* ≥ 30. *2.)* State the claim mathematically and verbally. Identify the null and alternative hypotheses. *3.)* Specify the level of significance. *4.)* Identify the degrees of freedom. *5.)* Determine the critical value(s). *6.)* Determine the rejection region(s). *7.)* Find the standardized test statistic and sketch the sampling distribution. *8.)* Make a decision to reject or fail to reject the null hypothesis. *9.)* Interpret the decision in the context of the original claim. *In Symbols:* *2.)* State *H₀* and *H*∨*a*. *3.)* Identify *α*. *4.)* d.f. = *n* - 1 *5.)* *Use Table 5 in Appendix B.* *7.)* *t* = (*x̄* - *μ*) / (*s*/√(*n*)) *8.)* If *t* is in the rejection region, then reject *H₀*. Otherwise, fail to reject *H₀*.

*Using Rejection Regions for a "z"-Test for a Mean "μ" ("σ" Known)* (p. 365)

*In Words:* *1.)* Verify that *σ* is known, the sample is random, and either the population is normally distributed or *n* ≥ 30. *2.)* State the claim mathematically and verbally. Identify the null and alternative hypotheses. *3.)* Specify the level of significance. *4.)* Determine the critical value(s). *5.)* Determine the rejection region(s). *6.)* Find the standardized test statistic and sketch the sampling distribution. *7.)* Make a decision to reject or fail to reject the null hypothesis. *8.)* Interpret the decision in the context of the original claim. *In Symbols:* *2.)* State *H₀* and *H*∨*a*. *3.)* Identify *α*. *4.)* *Use Table 4 in Appendix B.* *6.)* *z* = (*x̄* - *μ*) / (*σ*/√(*n*)) *7.)* If *z* is in the rejection region, then reject *H₀*. Otherwise, fail to reject *H₀*.

*Using "P"-Values for a "z"-Test for a Mean "μ" ("σ" Known)* (p. 365)

*In Words:* *1.)* Verify that *σ* is known, the sample is random, and either the population is normally distributed or *n* ≥ 30. *2.)* State the claim mathematically and verbally. Identify the null and alternative hypotheses. *3.)* Specify the level of significance. *4.)* Find the standardized test statistic. *5.)* Find the area that corresponds to *z*. *6.)* Find the *P*-value. ~ *[a]:* For a left-tailed test, *P* = (Area in left tail). ~ *[b]:* For a right-tailed test, *P* = (Area in right tail). ~ *[c]:* For a two-tailed test, *P* = 2×(Area in tail of standardized test statistic). *7.)* Make a decision to reject or fail to reject the null hypothesis. *8.)* Interpret the decision in the context of the original claim. *In Symbols:* *2.)* State *H₀* and *H*∨*a*. *3.)* Identify *α*. *4.)* *z* = (*x̄* - *μ*) / (*σ*/√(*n*)) *5.)* *Use Table 4 in Appendix B.* *7.)* If *P* ≤ *α*, then reject *H₀*. Otherwise, fail to reject *H₀*.

*Using the "z"-Test for a Proportion "p"* (p. 388)

*In Words:* *1.)* Verify that the sampling distribution of *p̂* can be approximated by a normal distribution. *2.)* State the claim mathematically and verbally. Identify the null and alternative hypotheses. *3.)* Specify the level of significance. *4.)* Determine the critical value(s). *5.)* Determine the rejection region(s). *6.)* Find the standardized test statistic and sketch the sampling distribution. *7.)* Make a decision to reject or fail to reject the null hypothesis. *8.)* Interpret the decision in the context of the original claim. *In Symbols:* *1.)* *np* ≥ 5, *nq* ≥ 5 *2.)* State *H₀* and *H*∨*a*. *3.)* Identify *α*. *4.)* *Use Table 4 in Appendix B.* *6.)* *z* = (*p̂* - *p*) / √((*p*×*q*)/(*n*). *7.)* If *z* is in the rejection region, then reject *H₀*. Otherwise, fail to reject *H₀*.

hypothesis test

A process that uses sample statistics to test a claim about the value of a population parameter. (p. 348)

statistical hypothesis

A statement about a population parameter. (p. 349)

null hypothesis (*H₀*)

A statistical hypothesis that contains a statement of equality, such as ≤, =, or ≥. (The symbol *H₀* is read as "H sub-zero" or "H naught".) (p. 349)

*"t" - Test for a Mean "μ"* (p. 379)

A statistical test for a population mean. The test statistic is the sample mean *x̄*. The standardized test statistic is: *t* = (*x̄* - *μ*) / (*s*/√(*n*)) ~~ Standardized test statistic for *μ* (*σ* unknown) where these conditions are met: *1.)* The sample is random. *2.)* At least one of the following is true: the population is normally distributed, or *n* ≥ 30. The degrees of freedom are d.f. = *n* - 1.

*"z"-Test for a Mean "μ"* (p. 365)

A statistical test for a population mean. The test statistic is the sample mean *x̄*. The standardized test statistic is: *z* = (*x̄* - *μ*) / (*σ*/√(*n*)) ~~ Standardized test statistic for *μ* (*σ* known) where these conditions are met: *1.)* The sample is random. *2.)* At least one of the following is true: the population is normally distributed, or *n* ≥ 30. Recall that *σ*/√*n* is the standard error of the mean, *σ∨x̄*.

*z* - Test for a Proportion *p* (p. 388)

A statistical test for population proportion. The *z* - test can be used when a binomial distribution is given such that *np* ≥ 5 and *nq* ≥ 5. The test statistic is the sample proportion *p̂* and the standardized test statistic is: *z* = (*p̂* - *μ*∨*p̂*) / *σ*∨*p̂* = (*p̂* - *p*) / √((*p*×*q*)/(*n*). ~~ Standardized test statistic for *p*

Finding the *P*-value for a Hypothesis Test (p. 363)

After determining the hypothesis test's standardized test statistic and the standardized test statistic's corresponding area, do one of the following to find the *P*-value: *[a]:* For a left-tailed test, *P* = (Area in left tail). *[b]:* For a right-tailed test, *P* = (Area in right tail). *[c]:* For a two-tailed test, *P* = 2×(Area in tail of standardized test statistic).

*(Y.T.I.): Exercise 5 (p. 358):* Your medical research team is investigating the mean cost of a​ 30-day supply of a certain heart medication. A pharmaceutical company thinks that the mean cost is more than $62. You want to support this claim. How would you write the null and alternative​ hypotheses? *H₀*: *___(1)___* *H*∨*a*: *___(2)___*

Correct Answer(s): *(1):* *μ ≤ 62* *(2):* *μ > 62* (Section 1)

*(Y.T.I.): Exercise 6 (p. 382):* A consumer group claims that the mean minimum time it takes for a sedan to travel a quarter mile is greater than 14.5 seconds. A random sample of 23 sedans has a mean minimum time to travel a quarter mile of 15.3 seconds and a standard deviation of 2.11 seconds. At *α* = 0.01 is there enough evidence to support the consumer​ group's claim? Complete parts 1 (a) through 5 (d) below. Assume the population is normally distributed. *Part 1 (a):* Identify the claim and state *H₀* and *H*∨*a*. (Type answers as either *integers* or *decimals*, but *DO NOT ROUND*.) *H₀*: *_(1)_* *_(2)_* *__(3)__* *H*∨*a*: *_(4)_* *_(5)_* *__(6)__* The claim is the *____(7)____* hypothesis. *Part 2 (b):* Use technology to find the​ *P* - value. Find the standardized test​ statistic, *t*. (Round answer to *two* decimal places.) *t* = *___* *Part 3 (b):* Obtain the *P* - value. (Round answer to *three* decimal places.) *P* = *____* *Part 4 (c):* Decide whether to reject or fail to reject the null hypothesis. *______(1)______* *H₀*, because the *P* - value is *_(2)_* than *α*. *Part 5 (d):* Interpret the decision in the context of the original claim. (Type answers as either *integers* or *decimals*, but *DO NOT ROUND*.) There *___(1)___* enough evidence at the *_(2)_%* level of significance to *___(3)___* the claim that the mean minimum time it takes for a sedan to travel a quarter mile is *______(4)______* *__(5)__* seconds.

Correct Answers: *Part 1 (a):* *(1):* *μ* *(2):* *≤* *(3):* *14.5* *(4):* *μ* *(5):* *>* *(6):* *14.5* *(7):* *alternative* *Part 2 (b):* *1.82* *Part 3 (b):* *0.041* *Part 4 (c):* *(1):* *Fail to reject* *(2):* *is* *Part 5 (d):* *(1):* *is not* *(2):* *1%* *(3):* *support* *(4):* *greater than* *(5):* *14.5* (Section 3)

*(Y.T.I.): Exercise 1 (p. 350):* For the statement​ below, write the claim as a mathematical statement. State the null and alternative hypotheses and identify which represents the claim. *A laptop manufacturer claims that the mean life of the battery for a certain model of laptop is more than 5 hours.* *Part 1:* Write the claim as a mathematical statement. A.) μ ≠ 5 B.) μ = 5 C.) μ ≤ 5 D.) μ > 5 E.) μ < 5 F.) μ ≥ 5 *Part 2:* Choose the correct null and alternative hypotheses below. A.) *H₀*: μ = 5 *H*∨*a*: μ ≠ 5 B.) *H₀*: μ ≠ 5 *H*∨*a*: μ = 5 C.) *H₀*: μ > 5 *H*∨*a*: μ ≤ 5 D.) *H₀*: μ ≥ 5 *H*∨*a*: μ < 5 E.) *H₀*: μ < 5 *H*∨*a*: μ ≥ 5 F.) *H₀*: μ ≤ 5 *H*∨*a*: μ > 5 *Part 3:* Identify which is the claim. A.) The alternative hypothesis *H*∨*a*: μ ≠ 5 is the claim. B.) The null hypothesis *H₀*: μ ≤ 5 is the claim. C.) The alternative hypothesis *H*∨*a*: μ > 5 is the claim. D.) The null hypothesis *H₀*: μ ≥ 5 is the claim E.) The alternative hypothesis *H*∨*a*: μ < 5 is the claim. F.) The null hypothesis *H₀*: μ = 5 is the claim.

Correct Answers: *Part 1:* D.) μ > 5 *Part 2:* F.) *H₀*: μ ≤ 5 *H*∨*a*: μ > 5 *Part 3:* C.) The alternative hypothesis *H*∨*a*: μ > 5 is the claim. (Section 1)

*(Y.T.I.): Exercise 1 (p. 389):* A medical researcher says that less than 86% of adults in a certain country think that healthy children should be required to be vaccinated. In a random sample of 400 adults in that​ country, 83% think that healthy children should be required to be vaccinated. At *α* = 0.01, is there enough evidence to support the​ researcher's claim? Complete parts 1 & 2 (a) through 6 (d & e) below. *Part 1 (a):* Identify the claim and state *H₀* and *H*∨*a*. Identify the claim in this scenario. Select the correct choice below and fill in the answer box to complete your choice. (Type answer(s) as either *integers* or *decimals*, but *DO NOT ROUND*.) A.) Less than *__%* of adults in the country think that healthy children should be required to be vaccinated. B.) *__%* of adults in the country think that healthy children should be required to be vaccinated. C.) The percentage of adults in the country who think that healthy children should be required to be vaccinated is not *__%*. D.) More than *__%* of adults in the country think that healthy children should be required to be vaccinated. *Part 2 (a):* Let p be the population proportion of​ successes, where a success is an adult in the country who thinks that healthy children should be required to be vaccinated. State *H₀* and *H*∨*a*. Select the correct choice below and fill in the answer boxes to complete your choice. (Round all answers to *two* decimal places.) A.) *H₀*: p = *___* *H*∨*a*: p ≠ *___* B.) *H₀*: p ≥ *___* *H*∨*a*: p < *___* C.) *H₀*: p ≠ *___* *H*∨*a*: p = *___* D.) *H₀*: p ≤ *___* *H*∨*a*: p > *___* E.) *H₀*: p > *___* *H*∨*a*: p ≤ *___* F.) *H₀*: p < *___* *H*∨*a*: p ≥ *___* *Part 3 (b):* Find the critical value(s) and identify the rejection region(s). (Round answer(s) to *two* decimal places, and use a *comma* to separate answers (if needed).) Identify the critical value(s) for this test. *z₀* = *___* *Part 4 (b):* Identify the rejection region(s). Select the correct choice below and fill in the answer​ box(es) to complete your choice. (Round answer(s) to *two* decimal places.) A.) The rejection regions are z < *___* and z > *___*. B.) The rejection region is *___* < z < *___*. C.) The rejection region is z > *___*. D.) The rejection region is z < *___*. *Part 5 (c):* Find the standardized test statistic *z*. (Round answer to *two* decimal places.) z = *___* *(Part 6 (d)):* Decide whether to reject or fail to reject the null hypothesis, and then *​(part 6 (e)):* interpret the decision in the context of the original claim. *________(1)________* the null hypothesis. There *____(2)____* enough evidence to *___(3)___* the​ researcher's claim.

Correct Answers: *Part 1 (a):* A.) Less than *86%* of adults in the country think that healthy children should be required to be vaccinated. *Part 2 (a):* B.) *H₀*: p ≥ *0.86* *H*∨*a*: p < *0.86* *Part 3 (b):* *-2.33* *Part 4 (b):* D.) The rejection region is z < *-2.33*. *Part 5 (c):* *-1.73* *Part 6 (d & e):* *(1):* *Fail to reject* *(2):* *is not* *(3):* *support* (Section 4)

*(Y.T.I.): Exercise 2 (p. 390):* A research center claims that 30% of adults in a certain country would travel into space on a commercial flight if they could afford it. In a random sample of 700 adults in that​ country, 34% say that they would travel into space on a commercial flight if they could afford it. At *α* = 0.05, is there enough evidence to reject the research​ center's claim? Complete parts 1 & 2 (a) through 5 (c & d) below. *(a):* Identify the claim and state *H₀* and *H*∨*a*. *Part 1 (a):* Identify the claim in this scenario. Select the correct choice below and fill in the answer box to complete your choice. (Type answer(s) as either *integers* or *decimals*, but *DO NOT ROUND*.) A.) The percentage adults in the country who would travel into space on a commercial flight if they could afford it is not *__%*. B.) *__%* of adults in the country would travel into space on a commercial flight if they could afford it. C.) At least *__%* of adults in the country would travel into space on a commercial flight if they could afford it. D.) No more than *__%* of adults in the country would travel into space on a commercial flight if they could afford it. *Part 2 (a):* Let p be the population proportion of​ successes, where a success is an adult in the country who thinks that healthy children should be required to be vaccinated. State *H₀* and *H*∨*a*. Select the correct choice below and fill in the answer boxes to complete your choice. (Round all answers to *two* decimal places.) A.) *H₀*: p > *___* *H*∨*a*: p ≤ *___* B.) *H₀*: p < *___* *H*∨*a*: p ≥ *___* C.) *H₀*: p ≠ *___* *H*∨*a*: p = *___* D.) *H₀*: p ≤ *___* *H*∨*a*: p > *___* E.) *H₀*: p ≥ *___* *H*∨*a*: p < *___* F.) *H₀*: p = *___* *H*∨*a*: p ≠ *___* *(b):* Use technology to find the *P* - value. *Part 3 (b):* Identify the standardized test statistic. (Round answer to *two* decimal places.) z = *___* *Part 4 (b):* Identify the *P* - value. (Round answer to *three* decimal places.) P = *____* *(Part 5 (c)):* Decide whether to reject or fail to reject the null hypothesis, and then *(part 5 (d)):* interpret the decision in the context of the original claim. *___(1)___* the null hypothesis. There *_(2)_* enough evidence to *___(3)___* the research​ center's claim.

Correct Answers: *Part 1 (a):* B.) *30%* of adults in the country would travel into space on a commercial flight if they could afford it. *Part 2 (a):* F.) *H₀*: p = *0.3* *H*∨*a*: p ≠ *0.3* *Part 3 (b):* *2.31* *Part 4 (b):* *0.021* *Part 5 (c & d):* *(1):* *Reject* *(2):* *is* *(3):* *reject* (Section 4)

*(Y.T.I.): Exercise 10 (p. 372):* A company that makes cola drinks states that the mean caffeine content per​ 12-ounce bottle of cola is 55 milligrams. You want to test this claim. During your​ tests, you find that a random sample of thirty​ 12-ounce bottles of cola has a mean caffeine content of 52.9 milligrams. Assume the population is normally distributed and the population standard deviation is 7.1 milligrams. At *α* = 0.09​, can you reject the​ company's claim? Complete parts​ 1 (a) through 6 (e) below. *Part 1 (a):* Identify *H₀* and *H*∨*a*. Choose the correct answer below. A.) *H₀*: μ = 52.9 *H*∨*a*: μ ≠ 52.9 B.) *H₀*: μ = 55 *H*∨*a*: μ ≠ 55 C.) *H₀*: μ ≤ 55 *H*∨*a*: μ > 55 D.) *H₀*: μ ≠ 52.9 *H*∨*a*: μ = 52.9 E.) *H₀*: μ ≠ 55 *H*∨*a*: μ = 55 F.) *H₀*: μ ≤ 52.9 *H*∨*a*: μ > 52.9 *Part 2 (b):* Find the critical value(s). Select the correct choice below and fill in the answer box within your choice. (Round answer(s) to *two* decimal places.) A.) The critical values are ± *___*. B.) The critical value is *___*. *Part 3 (b):* Identify the rejection​ region(s). Choose the correct answer below. (Since I don't have Quizlet+, I can't insert the images of the actual normal curves; ergo, I pasted their descriptions.) A.) A normal curve is over a horizontal axis labeled *z* from -4 to 4 in increments of 1, and is centered on 0. Vertical line segments extend to the left of and to the right of 0 from the horizontal axis to the curve. The area under the curve the left of the left line segment and to the right of the right lines segment is shaded, and is labeled *Reject "H₀"*. The area between the vertical line segments is labeled *Fail to reject "H₀"*. B.) A normal curve is over a horizontal axis labeled *z* from -4 to 4 in increments of 1, and is centered on 0. Vertical line segment extends to the left of 0 from the horizontal axis to the curve. The area under the curve to the left of the line segment is shaded, and is labeled *Reject "H₀"*. The area under the curve to the right of the line segment is labeled *Fail to reject "H₀"*. C.) A normal curve is over a horizontal axis labeled *z* from -4 to 4 in increments of 1, and is centered on 0. Vertical line segment extends to the right of 0 from the horizontal axis to the curve. The area under the curve to the right of the line segment is shaded, and is labeled *Reject "H₀"*. The area under the curve to the left of the line segment is labeled *Fail to reject "H₀"*. *Part 4 (c):* Find the standardized test statistic. (Round answer to *two* decimal places.) *z* = *___* *Part 5 (d):* Decide whether to reject or fail to reject the null hypothesis. A.) Since *z* is not in the rejection region, fail to reject the null hypothesis. B.) Since *z* is in the rejection region, reject the null hypothesis. C.) Since *z* is not in the rejection region, reject the null hypothesis. D.) Since *z* is in the rejection region, fail to reject the null hypothesis. *Part 6 (e):* Interpret the decision in the context of the original claim. At the 9% significance​ level, there *____(1)____* enough evidence to *__(2)__* the​ company's claim that the mean caffeine content per​ 12-ounce bottle of cola *_____(3)_____* *_(4)_* milligrams.

Correct Answers: *Part 1 (a):* B.) *H₀*: μ = 55 *H*∨*a*: μ ≠ 55 *Part 2 (b):* A.) The critical values are ± *1.70*. *Part 3 (b):* A.) A normal curve is over a horizontal axis labeled *z* from -4 to 4 in increments of 1, and is centered on 0. Vertical line segments extend to the left of and to the right of 0 from the horizontal axis to the curve. The area under the curve the left of the left line segment and to the right of the right lines segment is shaded, and is labeled *Reject "H₀"*. The area between the vertical line segments is labeled *Fail to reject "H₀"*. *Part 4 (c):* *-1.62* *Part 5 (d):* A.) Since *z* is not in the rejection region, fail to reject the null hypothesis. *Part 6 (e):* *(1):* *is not* *(2):* *reject* *(3):* *is equal to* *(4):* *55* (Section 2)

*(Y.T.I.): Exercise 4 (p. 380):* A credit card company claims that the mean credit card debt for individuals is greater than $4,900. You want to test this claim. You find that a random sample of 27 cardholders has a mean credit card balance of $5,151 and a standard deviation of $575. At *α* = 0.10, can you support the​ claim? Complete parts 1 (a) through 6 (e) below. Assume the population is normally distributed. *Part 1 (a):* Write the claim mathematically and identify *H₀* and *H*∨*a*. Which of the following correctly states *H₀* and *H*∨*a*? A.) *H₀*: μ ≥ $4,900 *H*∨*a*: μ < $4,900 B.) *H₀*: μ = $4,900 *H*∨*a*: μ > $4,900 C.) *H₀*: μ ≤ $4,900 *H*∨*a*: μ > $4,900 D.) *H₀*: μ = $4,900 *H*∨*a*: μ ≠ $4,900 E.) *H₀*: μ > $4,900 *H*∨*a*: μ ≤ $4,900 F.) *H₀*: μ > $4,900 *H*∨*a*: μ ≤ $4,900 *Part 2 (b):* Find the critical value(s) and identify the rejection region(s). (Round answer(s) to *three* decimal places, and use a *comma* to separate answers (if needed).) What is(are) the critical value(s), *t₀​*? *t₀​* = *____* *Part 3 (b):* Determine the rejection region(s). Select the correct choice below and fill in the answer​ box(es) within your choice. (Round answer(s) to *three* decimal places.) A.) t < *____* B.) *____* < t < *____* C.) t < *____* and t > *____* D.) t > *____* *Part 4 (c):* Find the standardized test statistic *t*. (Round answer to *two* decimal places.) *t* = *___* *Part 5 (d):* Decide whether to reject or fail to reject the null hypothesis. A.) Reject *H₀*, because the test statistic is in the rejection region. B.) Fail to reject *H₀*, because the test statistic is not in the rejection region. C.) Fail to reject *H₀*, because the test statistic is in the rejection region. D.) Reject *H₀*, because the test statistic is not in the rejection region. *Part 6 (e):* Interpret the decision in the context of the original claim. A.) At the 10% level of​ significance, there is sufficient evidence to support the claim that the mean credit card debt is less than $4,900. B.) At the 10% level of​ significance, there is not sufficient evidence to support the claim that the mean credit card debt is less than $4,900. C.) At the 10% level of​ significance, there is not sufficient evidence to support the claim that the mean credit card debt is greater than $4,900. D.) At the 10% level of​ significance, there is sufficient evidence to support the claim that the mean credit card debt is greater than $4,900.

Correct Answers: *Part 1 (a):* C.) *H₀*: μ ≤ $4,900 *H*∨*a*: μ > $4,900 *Part 2 (b):* *1.315* *Part 3 (b):* D.) t > *1.315* *Part 4 (c):* *2.27* *Part 5 (d):* A.) Reject *H₀*, because the test statistic is in the rejection region. *Part 6 (e):* D.) At the 10% level of​ significance, there is sufficient evidence to support the claim that the mean credit card debt is greater than $4,900. (Section 3)

*(Y.T.I.): Exercise 4 (p. 366):* A random sample of 88 eighth grade​ students' scores on a national mathematics assessment test has a mean score of 283. 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 280. Assume that the population standard deviation is 37. At *α* = 0.05, is there enough evidence to support the​ administrator's claim? Complete parts 1 (a) through 5 (e) below. *Part 1 (a):* Write the claim mathematically and identify *H₀* and *H*∨*a*. Choose the correct answer below. A.) *H₀*: μ = 280 ​(claim) *H*∨*a*: μ > 280 B.) *H₀*: μ = 280 *H*∨*a*: μ > 280 ​(claim) C.) *H₀*: μ ≤ 280 *H*∨*a*: μ > 280 ​(claim) D.) *H₀*: μ ≥ 280 ​(claim) *H*∨*a*: μ < 280 E.) *H₀*: μ < 280 *H*∨*a*: μ ≥ 280 ​(claim) F.) *H₀*: μ ≤ 280 ​(claim) *H*∨*a*: μ > 280 *Part 2 (b):* Find the standardized test statistic *z*. (Round answer to *two* decimal places.) z = *___* *Part 3 (c):* Find the *P* - value. (Round answer to *three* decimal places.) *P* - value = *____* *Part 4 (d):* Decide whether to reject or fail to reject the null hypothesis. A.) Fail to reject *H₀*. B.) Reject *H₀*. *Part 5 (e):* Interpret your decision in the context of the original claim. At the 5% significance​ level, there *____(1)____* enough evidence to *___(2)___* the​ administrator's claim that the mean score for the​ state's eighth graders on the exam is more than 280.

Correct Answers: *Part 1 (a):* C.) *H₀*: μ ≤ 280 *H*∨*a*: μ > 280 (claim) *Part 2 (b):* *0.76* *Part 3 (c):* *0.224* *Part 4 (d):* A.) Fail to reject *H₀*. *Part 5 (e):* *(1):* *is not* *(2):* *support* (Section 2)

*(Y.T.I.): Exercise 5 (p. 381):* A used car dealer says that the mean price of a​ three-year-old sports utility vehicle is ​$23,000. You suspect this claim is incorrect and find that a random sample of 21 similar vehicles has a mean price of ​$23,738 and a standard deviation of ​$1,978. Is there enough evidence to reject the claim at *α* = 0.05​? Complete parts 1 (a) through 6 (e) below. Assume the population is normally distributed. *Part 1 (a):* Write the claim mathematically and identify *H₀* and *H*∨*a*. Which of the following correctly states *H₀* and *H*∨*a*? A.) *H₀*: μ = $23,000 *H*∨*a*: μ < $23,000 B.) *H₀*: μ = $23,000 *H*∨*a*: μ > $23,000 C.) *H₀*: μ ≥ $23,000 *H*∨*a*: μ < $23,000 D.) *H₀*: μ > $23,000 *H*∨*a*: μ ≤ $23,000 E.) *H₀*: μ ≠ $23,000 *H*∨*a*: μ = $23,000 F.) *H₀*: μ = $23,000 *H*∨*a*: μ ≠ $23,000 *Part 2 (b):* Find the critical value(s) and identify the rejection region(s). (Round answer(s) to *three* decimal places, and use a *comma* to separate answers (if needed).) What is(are) the critical value(s), *t₀​*? *t₀​* = *__(1)__*, *__(2)__* *Part 3 (b):* Determine the rejection region(s). Select the correct choice below and fill in the answer​ box(es) within your choice. (Round answer(s) to *three* decimal places.) A.) t < *____* B.) *____* < t < *____* C.) t > *____* D.) t < *____* and t > *____* *Part 4 (c):* Find the standardized test statistic *t*. (Round answer to *two* decimal places.) *t* = *___* *Part 5 (d):* Decide whether to reject or fail to reject the null hypothesis. A.) Reject *H₀*, because the test statistic is not in the rejection region(s). B.) Fail to reject *H₀*, because the test statistic is in the rejection region(s). C.) Reject *H₀*, because the test statistic is in the rejection region(s). D.) Fail to reject *H₀*, because the test statistic is not in the rejection region(s). *Part 6 (e):* Interpret the decision in the context of the original claim. A.) At the 5% level of​ significance, there is sufficient evidence to reject the claim that the mean price is not $23,000. B.) At the 5% level of​ significance, there is sufficient evidence to reject the claim that the mean price is $23,000. C.) At the 5% level of​ significance, there is not sufficient evidence to reject the claim that the mean price is not $23,000. D.) At the 5% level of​ significance, there is not sufficient evidence to reject the claim that the mean price is $23,000.

Correct Answers: *Part 1 (a):* F.) *H₀*: μ = $23,000 *H*∨*a*: μ ≠ $23,000 *Part 2 (b):* *(1):* *-2.086,* *(2):* *2.086* *Part 3 (b):* D.) t < *-2.086* and t > *2.086* *Part 4 (c):* *1.71* *Part 5 (d):* D.) Fail to reject *H₀*, because the test statistic is not in the rejection region(s). *Part 6 (e):* D.) At the 5% level of​ significance, there is not sufficient evidence to reject the claim that the mean price is $23,000. (Section 3)

*(Y.T.I.): Exercise 8 (p. 369):* Find the critical​ value(s) and rejection​ region(s) for the type of​ *z* - test with level of significance *α*. Include a graph with your answer. *Two ​- tailed test, α = 0.03* *Part 1:* The critical​ value(s) is/are z = *__(1)__*, *__(2)__*. (Round answer(s) to *two* decimal places, and use a *comma* to separate answers (if needed).) *Part 2:* Select the correct choice below​ and, if necessary, fill in the answer box to complete your choice. (Round answer(s) to *two* decimal places.) A.) The rejection region is *z* < *___*. B.) The rejection regions are *z* < *___* and *z* > *___*. C.) The rejection region is *z* > *___*. *Part 3:* Choose the correct graph of the rejection region below. (Since I don't have Quizlet+, I can't insert the images of the actual normal curves; ergo, I pasted their descriptions.) A.) A normal curve is over a horizontal axis, and is centered at 0. Vertical line segments extend from the horizontal axis to the curve at 0 and *z*, where *z* is to the right of 0. The area under the curve to the right of *z* is shaded. B.) A normal curve is over a horizontal axis, and is centered at 0. Vertical line segments extend from the horizontal axis to the curve at *-z*, 0, and *z*, where *-z* is to the left of 0, and *z* is to the right of 0. The area under the curve to the left of *-z* and to the right of *z* is shaded. C.) A normal curve is over a horizontal axis, and is centered at 0. Vertical line segments extend from the horizontal axis to the curve at 0 and *z*, where *z* is to the right of 0. The area under the curve to the left of *z* is shaded. D.) A normal curve is over a horizontal axis, and is centered at 0. Vertical line segments extend from the horizontal axis to the curve at *-z*, 0, and *z*, where *-z* is to the left of 0, and *z* is to the right of 0. The area under the curve between *-z* and *z* is shaded.

Correct Answers: *Part 1:* *(1):* *-2.17* *(2):* *2.17* *Part 2:* B.) The rejection regions are *z* < *-2.17* and *z* > *2.17*. *Part 3:* B.) A normal curve is over a horizontal axis, and is centered at 0. Vertical line segments extend from the horizontal axis to the curve at *-z*, 0, and *z*, where *-z* is to the left of 0, and *z* is to the right of 0. The area under the curve to the left of *-z* and to the right of *z* is shaded. (Section 2)

*(Y.T.I.): Exercise 4 (p. 356):* 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 more than 56 days.* *Part 1:* Does the claim represent the null hypothesis or the alternative​ hypothesis? Since the claim *_______(1)_______* a statement of​ equality, it represents the *_____(2)_____* hypothesis. *Part 2 (a):* How should you interpret a decision that rejects the null​ hypothesis? There is *____(1)____* evidence to *____(2)____* the claim that the mean incubation period for the eggs of a species of bird is more than 56 days. *Part 3 (b):* How should you interpret a decision that fails to reject the null​ hypothesis? There is *_____(1)_____* evidence to *____(2)____* the claim that the mean incubation period for the eggs of a species of bird is more than 56 days.

Correct Answers: *Part 1:* *(1):* *does not contain* *(2):* *alternative* *Part 2 (a):* *(1):* *sufficient* *(2):* *support* *Part 3 (b):* *(1):* *insufficient* *(2):* *support* (Section 1)

*(Y.T.I.): Exercise 1 (p. 377):* 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" = 0.10*, *n = 20* *(I didn't copy the accompanying "t" - Distribution Table because it's too big.)* *Part 1:* The critical​ value(s) is/are *____*. (Round answer(s) to the *nearest thousandth* (*three* decimal places), and use a *comma* to separate answers (if needed).) *Part 2:* Determine the rejection region(s). Select the correct choice below and fill in the answer​ box(es) within your choice. (Round answer(s) to the *nearest thousandth* (*three* decimal places).) A.) t < *____* B.) *____* < t < *____* C.) t > *____* D.) t < *____* and t > *____*

Correct Answers: *Part 1:* *-1.328* *Part 2:* A.) t < *-1.328* (Section 3)

*(Y.T.I.): Exercise 2 (p. 364):* Find the​ *P* - value for a​ left-tailed hypothesis test with a test statistic of *z* = −1.31. (Round answer to *four* decimal places.) Decide whether to reject *H₀* if the level of significance is *α* = 0.10. *Part 1:* *P* - value = *_____* *Part 2:* State your conclusion. Choose the correct answer below. A.) Since *P* > *α*, fail to reject *H₀*. B.) Since *P* ≤ *α*, reject *H₀*. C.) Since *P* ≤ *α*, fail to reject *H₀*. D.) Since *P* > *α*, reject *H₀*.

Correct Answers: *Part 1:* *0.0951* *Part 2:* B.) Since *P* ≤ *α*, reject *H₀*. (Section 2)

*(Y.T.I.): Exercise 2 (p. 351):* Describe type I and type II errors for a hypothesis test of the indicated claim. *A furniture store claims that no more than 80​% of its new customers will return to buy their next piece of furniture.* *Part 1:* Describe the type I error. Choose the correct answer below. A.) A type I error will occur when the actual proportion of new customers who return to buy their next piece of furniture is no more than 0.80, but you reject *H₀*: p ≤ 0.80. B.) A type I error will occur when the actual proportion of new customers who return to buy their next piece of furniture is no more than 0.80, but you fail to reject *H₀*: p ≤ 0.80. C.) A type I error will occur when the actual proportion of new customers who return to buy their next piece of furniture is at least 0.80, but you reject *H₀*: p ≥ 0.80. D.) A type I error will occur when the actual proportion of new customers who return to buy their next piece of furniture is at least 0.80, but you fail to reject *H₀*: p ≥ 0.80. *Part 2:* Describe the type II error. Choose the correct answer below. A.) A type II error will occur when the actual proportion of new customers who return to buy their next piece of furniture is more than 0.80, but you fail to reject *H₀*: p ≤ 0.80. B.) A type II error will occur when the actual proportion of new customers who return to buy their next piece of furniture is less than 0.80, but you reject *H₀*: p ≥ 0.80. C.) A type II error will occur when the actual proportion of new customers who return to buy their next piece of furniture is less than 0.80, but you fail to reject *H₀*: p ≤ 0.80. D.) A type II error will occur when the actual proportion of new customers who return to buy their next piece of furniture is mor than 0.80, but you reject *H₀*: p ≥ 0.80.

Correct Answers: *Part 1:* A.) A type I error will occur when the actual proportion of new customers who return to buy their next piece of furniture is no more than 0.80, but you reject *H₀*: p ≤ 0.80. *Part 2:* A.) A type II error will occur when the actual proportion of new customers who return to buy their next piece of furniture is more than 0.80, but you fail to reject *H₀*: p ≤ 0.80. (Section 1)

Section 4

Hypothesis Test for Proportions

Section 2

Hypothesis Testing for the Mean (*σ* Known)

Section 3

Hypothesis Testing for the Mean (*σ* Unknown)

level of significance

In a hypothesis test, the maximum allowable probability of making a type I error. It is denoted by *α*, the lowercase Greek letter alpha. (The probability of a type II error is denoted by *β*, the lowercase Greek letter beta.) (p. 353)

Section 1

Introduction to Hypothesis Testing

Type II error

Occurs if the null hypothesis is not rejected when it is false. (p. 351)

Type I error

Occurs if the null hypothesis is rejected when it is true. (p. 351)

alternative hypothesis (*H*∨*a*)

The complement of the null hypothesis. It is a statement that must be true if *H₀* is false and contains a statement of strict inequality, such as >, ≠, or <. (The symbol *H*∨*a* is read as "H sub-a".) (p. 349)

*P* - value (probability value)

The probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined by the sample data. (p. 353)

rejection region

The range of values for which the null hypothesis is not probable. If a standardized test statistic falls in this region, then the null hypothesis is rejected. A critical value *"z"₀* separates the rejection region from the nonrejection region. Also called the *critical region*. (p. 368)

Decision Rule Based on *P*-Value (p. 356)

To use a *P*-value make a decision in a hypothesis test, compare the *P*-value with *α*. *1.)* If *P* ≤ *α*, then reject *H₀*. *2.)* If *P* > *α*, then fail to reject *H₀*.

*Decision Rule Based on Rejection Region* (p. 370)

To use a rejection region to conduct a hypothesis test, calculate the standardized test statistic *z*. If the standardized test statistic: *1.)* is in the rejection region, then reject *H₀*. *2.)* is *not* in the rejection region, then fail to reject *H₀*.

right-tailed test (p. 354)

Type of hypothesis test that occurs if the alternative hypothesis *H*∨*a* contains the greater-than inequality symbol (>). (Since I don't have Quizlet+, I can't insert the image of the actual normal curve; ergo, I pasted (some of) the information.) *H₀*: *μ* ≤ *k* *H*∨*a*: *μ* > *k* (*P* is the area to the right of the standardized test statistic.)

left-tailed test (p. 354)

Type of hypothesis test that occurs if the alternative hypothesis *H*∨*a* contains the less-than inequality symbol (<). (Since I don't have Quizlet+, I can't insert the image of the actual normal curve; ergo, I pasted (some of) the information.) *H₀*: *μ* ≥ *k* *H*∨*a*: *μ* < *k* (*P* is the area to the left of the standardized test statistic.)

two-tailed test (p. 354)

Type of hypothesis test that occurs if the alternative hypothesis *H*∨*a* contains the not-equal-to symbol (≠). In a *two-tailed test*, each tail has an area of *½*×*P*. (Since I don't have Quizlet+, I can't insert the image of the actual normal curve; ergo, I pasted (some of) the information.) *H₀*: *μ* = *k* *H*∨*a*: *μ* ≠ *k* ~~ *[a]:* The area to the *left* of the *negative* standardized test statistic is *½*×*P*. ~~ *[b]:* The area to the *right* of the *positive* standardized test statistic is *½*×*P*.