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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 8 - Hypothesis Testing with Two Samples문제 8.1.15
8장, 문제 8.1.15

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.
Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

검증된 단계별 안내
1
Step 1: Identify the claim and state the null hypothesis (Ho) and alternative hypothesis (Ha). The claim is that the mean braking distances for compact SUVs and midsize SUVs are different. The null hypothesis (Ho) states that the mean braking distances are equal: Ho: μ1 = μ2. The alternative hypothesis (Ha) states that the mean braking distances are different: Ha: μ1 ≠ μ2.
Step 2: Determine the critical value(s) and rejection region(s). Since the test is two-tailed (due to the claim of 'difference'), use the significance level α = 0.10 to find the critical z-values. The rejection regions will be in the tails of the standard normal distribution, corresponding to the critical z-values.
Step 3: Calculate the standardized test statistic z. Use the formula for the z-test for two independent means: z=(x1-x2)σ12n1+σ22n2, where x1 and x2 are the sample means, σ1 and σ2 are the population standard deviations, and n1 and n2 are the sample sizes.
Step 4: Compare the calculated z-value to the critical z-values. If the calculated z-value falls within the rejection region (outside the range defined by the critical z-values), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is sufficient evidence to support the claim that the mean braking distances are different for compact SUVs and midsize SUVs. If the null hypothesis is not rejected, conclude that there is insufficient evidence to support the claim.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. In this context, the engineer will test whether the mean braking distances of the two SUV categories differ significantly.
추천 영상:
가이드 코스
06:21
Step 1: Write Hypotheses

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to the rejection of H0. In this case, with α=0.10, the engineer will identify the critical z-value to assess whether the observed test statistic falls within this region.
추천 영상:
05:50
Critical Values: t-Distribution

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an observed sample mean is from the hypothesized population mean under the null hypothesis. It is calculated using the means, standard deviations, and sample sizes of the groups being compared. In this scenario, the engineer will compute the z-value to evaluate the difference in mean braking distances between the compact and midsize SUVs, facilitating the decision to reject or fail to reject the null hypothesis.
추천 영상:
가이드 코스
06:34
Step 2: Calculate Test Statistic
관련 실천
교과서 질문

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

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교과서 질문

[APPLET] Teaching Methods

A new method of teaching reading is being tested on third grade students. A group of third grade students is taught using the new curriculum. A control group of third grade students is taught using the old curriculum. The reading test scores for the two groups are shown in the back-to-back stem-and-leaf plot.

At , α=0.10 is there enough evidence to support the claim that the new method of teaching reading produces higher reading test scores than the old method does? Assume the population variances are equal.

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교과서 질문

Blue Crabs A marine researcher claims that the stomachs of blue crabs from one location contain more fish than the stomachs of blue crabs from another location. The stomach contents of a sample of 25 blue crabs from Location A contain a mean of 320 milligrams of fish and a standard deviation of 60 milligrams. The stomach contents of a sample of 15 blue crabs from Location B contain a mean of 280 milligrams of fish and a standard deviation of 80 milligrams. At , α= 0.01can you support the marine researcher’s claim? Assume the population variances are equal.

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교과서 질문

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.03

Population statistics:σ1=136 and σ2=215

Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156

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교과서 질문

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

ACT English and Reading Scores The mean ACT English score for 120 high school students is 19.9. Assume the population standard deviation is 7.2. The mean ACT reading score for 150 high school students is 21.2. Assume the population standard deviation is 7.1. At α=0.10, can you support the claim that ACT reading scores are higher than ACT English scores? (Source: ACT, Inc.)

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교과서 질문

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal and (b) not equal.

Ha:μ1>μ2 , α=0.01 , n1=12 , n2=15

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