Stats 8.1 & 8.2

Critical Values Z test

(=): ½ × α; 1- ½ × α. (<): α (>):1 - α standard normal table

Use the given information to answer parts​ (a) through​ (d). H₀: µ₁ = µ₂, α = 0.20 Sample statistics: xbar₁ = 30.2, s₁ = 3.7, n₁ = 18. xbar₂ = 31.7, s₂ = 2.3, n₂ = 8. Assume σ₁² = σ₂². -t₀ = -1.318, t₀ = 1.318. (a) Find the test statistic. ​(b) Find the standardized test statistic. ​(c) Decide whether the standardized test statistic is in the rejection region. (d) Decide whether you should reject or fail to reject the null hypothesis.

(a) -1.5 (b) t = -1.053 (c) The standardized test statistic, t, is not in the rejection region. (d) Fail to reject the null hypothesis.

Use the given information to answer parts​ (a) through​ (d). H₀: µ₁ = µ₂, α = 0.10 Sample statistics: xbar₁ = 2220, s₁ = 151, n₁ = 6. xbar₂ = 2311, s₂ = 37, n₂ = 8. Assume σ₁² ≠ σ₂². t₀ = 1.476. (a) Find the test statistic. ​(b) Find the standardized test statistic. ​(c) Decide whether the standardized test statistic is in the rejection region. (d) Decide whether you should reject or fail to reject the null hypothesis.

(a) -91 (b) t = -1.444 (c)The standardized test statistic, t, is not in the rejection region. (d) Fail to reject the null hypothesis.

A pet association claims that the mean annual cost of food for dogs and cats are the same. The results for samples for the two types of pets are shown below. At α = 0.05​, can you reject the pet​ association's claim? Assume the population variances are not equal. Assume the samples are random and​ independent, and the populations are normally distributed. Complete parts​ (a) through​ (e) below. Dogs: xbar₁ = $212, s₁ = $38, n₁ = 17. Cats: xbar₁ = $177, s₁ = $34, n₁ = 14. ​(a) Identify the claim and state H₀ and Ha. Which is the correct claim? What are H₀ and Ha? Which hypothesis is the claim? ​(b) Find the critical​ value(s) and identify the rejection​ region(s). ​(c) Find the standardized test statistic. (d) Decide whether to reject or fail to reject the null hypothesis. ​(e) Interpret the decision in the context of the original claim.

(a) Claim: "The mean annual costs for food for dogs and cats are equal." H₀ and Ha: The null hypothesis, H₀, is µ₁ = µ₂. The alternative hypothesis, Ha, is µ₁ ≠ µ₂. Hypothesis: The null hypothesis, H₀. (b) Critical value(s): -2.160, 2.160. Rejection region(s): t < -t₀, t > t₀. (c) t = 2.704 (d) Reject the null hypothesis. (e) At the 5% significance level, there is enough evidence to reject the claim that the mean annual costs of food for dogs and cats are the same.

A researcher claims that the stomachs of blue crabs from Location A contain more fish than the stomachs of blue crabs from Location B. The stomach contents of a sample of 15 blue crabs from Location A contain a mean of 195 milligrams of fish and a standard deviation of 37 milligrams. The stomach contents of a sample of 10 blue crabs from Location B contain a mean of 183 milligrams of fish and a standard deviation of 43 milligrams. At α = 0.01​, can you support the​ researcher's claim? Assume the population variances are equal. Complete parts​ (a) through​ (d) below. (a) Identify the null and alternative hypotheses. ​(b) Find the standardized test statistic for µ₁ - µ₂. ​(c) Calculate the​ P-value. ​(d) State the conclusion.

(a) H₀: µ₁ - µ₂ ≤ 0; Ha: µ₁ - µ₂ > 0 (b) t = .745 (c) P = .2319 (d) Fail to reject H₀. There is not enough evidence at the 1% level of significance to support the researcher's claim.

A group says that the mean daily lodging cost for a family traveling in Region 1 is the same as in Region 2. The mean daily lodging cost for 34 families traveling in Region 1 is ​$120 with a population standard deviation of ​$28. The mean daily lodging cost for 34 families traveling in Region 2 is ​$128 with a population standard deviation of ​$20. At α = 0.10, is there enough evidence to reject the​ group's claim? Complete parts​ (a) through​ (e). (a) Identify the claim and state H₀ and Ha. What is the claim? What are H₀ and Ha? ​(b) Find the critical​ value(s) and identify the rejection​ region(s). What is/are the rejection region(s)? ​(c) Find the standardized test statistic z for µ₁ - µ₂. (d) Decide whether to reject or fail to reject the null hypothesis. Choose the correct answer below. ​(e) Interpret the decision in the context of the original claim.

(a) The mean daily lodging costs in Region 1 and Region 2 are equal. H₀: µ₁ = µ₂ Ha: µ₁ ≠ µ₂ (b) The critical value(s) is/are -1.645, 1.645. z < -1.645, z > 1.645 z = -1.356 (d) Fail to reject H₀. The standardized test statistic is not in the rejection region. (e) At the 10% significance level, there is insufficient evidence to reject the claim that the mean daily lodging costs in Region 1 is equal to the mean daily lodging costs in Region 2.

An energy company wants to choose between two regions in a state to install​ energy-producing wind turbines. A researcher claims that the wind speed in Region A is less than the wind speed in Region B. To test the​ regions, the average wind speed is calculated for 90 days in each region. The mean wind speed in Region A is 13.7 miles per hour. Assume the population standard deviation is 2.8 miles per hour. The mean wind speed in Region B is 15.3 miles per hour. Assume the population standard deviation is 3.2 miles per hour. At α = 0.05​, can the company support the​ researcher's claim? Complete parts​ (a) through​ (d) below. (a) Identify the claim and state H₀ and Ha. What is the​ claim? Let Region A be sample 1 and let Region B be sample 2. Identify H₀ and Ha. (b) Find the critical​ value(s) and identify the rejection region. What is the rejection​ region? Select the correct choice below and fill in the answer​ box(es) within your choice. ​(c) Find the standardized test statistic z. ​(d) Decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim.

(a) The wind speed in Region A is less than the wind speed in Region B. H₀: µ₁ ≥ µ₂ Ha: µ₁ < µ₂ (b) The critical​ value(s) is/are z₀ = -1.645. z < -1.645 (c) z = -3.57 (d) Reject H₀. There is enough evidence at the 5% level of significance to support the researcher's claim that the wind speed in Region A is less than the wind speed in Region B.

What conditions are necessary in order to use the​ z-test to test the difference between two population​ means?

- Each population has a normal distribution with a known standard deviation. - The samples must be independent. - The samples must be randomly selected.

Test the claim about the difference between two population means mu 1 and mu 2 at the level of significance alpha. Assume the samples are random and​ independent, and the populations are normally distributed. ​Claim: µ₁= µ₂​; α = 0.01 Population​ statistics: σ₁ = 3.2​, σ₂ = 1.4 Sample​ statistics: xbar₁ = 17, n₁ = 28​, xbar₂ = 15, n₂ = 27 Determine the alternative hypothesis. Determine the standardized test statistic. Determine the​ P-value. What is the proper​ decision?

Ha​: µ₁ ≠ µ₂ z = 3.02 P-value = .003 Reject H₀. There is enough evidence at the 1% level of significance to reject the claim.

Reject or Fail to Reject

If the​ P-value is less than or equal to the level of​ significance, reject H₀. ​Otherwise, fail to reject H₀.

What conditions are necessary in order to use a t​-test to test the differences between two population​ means?

The population standard deviations are unknown. The samples are randomly selected and independent. The populations are normally distributed or each sample size is at least 30.Th

Classify the two given samples as independent or dependent. Sample​ 1: The average test scores of 47 math students Sample​ 2: The average test scores of the same 47 math students after being tutored for two months

The two given samples are dependent because the same students were sampled.

Classify the two given samples as independent or dependent. Sample​ 1: The weights of 38 oranges grown in California Sample​ 2: The weights of 44 oranges grown in Florida

The two given samples are independent because different oranges were sampled.

Classify the two given samples as independent or dependent. Sample​ 1: The average time for 37 professional skiers to finish a race Sample​ 2: The average time for 43 amateur skiers to finish a race

The two given samples are independent because different skiers were sampled.

Classify the two given samples as independent or dependent. Sample​ 1: The gas mileage for 39 trucks Sample​ 2: The gas mileage for 43 cars

The two given samples are independent because different vehicles were sampled.

Critical values T test

equal pop variance(σ): n₁ + n₂ - 2. not equal pop variance(σ): minimum of n₁ - 1 or n₂ - 1. T distribution chart

Support or Reject

null hypothesis claim: reject alternative hypothesis claim: support

Use the​ t-distribution table to find the critical​ value(s) for the indicated alternative​ hypotheses, level of significance α, and sample sizes n₁ and n₂. Assume that the samples are​ independent, normal, and random. Answer parts​ (a) and​ (b). Ha: µ₁ ≠ µ₂, α = 0.10, n₁ = 17, n₂ = 8 ​(a) Find the critical​ value(s) assuming that the population variances are equal. (b) Find the critical​ value(s) assuming that the population variances are not equal.

population variance equal: n₁ + n₂ - 2 population variance not equal: minimum value of n₁ - 1, n₂ - 1. (<) = (-), (>) = (+), (=) = (±). (a) -1.717, 1.714. (b) -1.895, 1.895.

pool or no pool

population variances(σ) equal: pool population variances(σ) not equal: no pool

Test Statistic

xbar₁ - xbar₂

Standardized Test Statistic

z = (xbar₁ - xbar₂) - (µ¹- µ₂)/√σ₁²/n₁ + σ₂²/n₂ Calculator: STAT; TESTS; 2-SampZTest when σ is known; 2-SampTTest when σ is unnown.

A safety engineer records the braking distances of two types of tires. Each randomly selected sample has 35 tires. The results of the tests are shown in the table. At α = 0.10​, can the engineer support the claim that the mean braking distance is different for the two types of​ tires? Assume the samples are randomly selected and that the samples are independent. Complete parts​ (a) through​ (e). Type A: xbar₁ = 40 feet σ₁ = 4.5 feet Type B: xbar₂ = 41 feet σ₂ = 4.2 feet ​(a) Identify the claim and state H₀ and Ha. What is the​ claim? What are H₀ and Ha​? ​(b) Find the critical​ value(s) and identify the rejection​ region(s). ​(c) Find the standardized test statistic z for µ₁ - µ₂. (d) Decide whether to reject or fail to reject the null hypothesis. Choose the correct answer below. ​(e) Interpret the decision in the context of the original claim.

​(a) The mean braking distance is different for the two types of tires. H₀: µ₁ = µ₂ Ha: µ₁ ≠ µ₂ ​(b)The critical​ value(s) is/are -1.645, 1.645. z < -1.645​, z > 1.645. (c) z = -.961 (d) Fail to reject H₀. The standardized test statistic does not fall in the rejection region. (e) At the 10​% significance​ level, there is insufficient evidence to support the claim that the mean braking distance for Type A tires is different from the one for Type B tires.

For the given​ data, (a) find the test​ statistic, (b) find the standardized test​ statistic, (c) decide whether the standardized test statistic is in the rejection​ region, and​ (d) decide whether you should reject or fail to reject the null hypothesis. The samples are random and independent. Claim: µ₁ < µ₂, α = 0.01. Sample statistics: xbar₁= 1255, n₁ = 45, xbar₂ 1205, n₂ = 65. Population statistics: σ₁ = 70 and σ₂ = 105.

​(a) The test statistic for µ₁ - µ₂ is 50. ​(b) The standardized test statistic for µ₁ - µ₂ is 3. ​(c) Is the standardized test statistic in the rejection​ region? - No (d) Should you reject or fail to reject the null​ hypothesis? - H₀: µ₁ ≥ µ₂; Ha: µ₁< µ₂. Fail to reject H₀. At the 1% significance level, there is not enough evidence to support the claim.