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

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

문제 8.R.16

8장, 문제 8.R.16

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1> μ2; α=0.10. Assume (σ1)^2 ≠ (σ2)^2

Sample statistics: x̅1= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6

검증된 단계별 안내

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) is μ₁ ≤ μ₂, and the alternative hypothesis (H₁) is μ₁ > μ₂. This is a one-tailed test since the claim specifies μ₁ > μ₂.

Step 2: Determine the test statistic formula for comparing two population means when the population variances are unequal. Use the formula:

\frac{({x̅}_{1} - {x̅}_{2})}{\sqrt{s_{1}^{2} + s_{2}^{2}}}

, where x̅₁ and x̅₂ are the sample means, s₁² and s₂² are the sample variances, and n₁ and n₂ are the sample sizes.

Step 3: Calculate the degrees of freedom using the formula for unequal variances:

\frac{(s_{1}^{2} + s_{2}^{2}) 2}{s_{1}^{4} + s_{2}^{4}}

. This will give an approximate degrees of freedom for the t-distribution.

Step 4: Use the calculated test statistic and degrees of freedom to find the critical value from the t-distribution table at the significance level α = 0.10 for a one-tailed test. Compare the test statistic to the critical value to determine whether to reject or fail to reject the null hypothesis.

Step 5: Interpret the results. If the test statistic exceeds the critical value, reject the null hypothesis and conclude that there is sufficient evidence to support the claim μ₁ > μ₂. Otherwise, fail to reject the null hypothesis and conclude that there is insufficient evidence to support the claim.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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 a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the claim is that the mean of population 1 (μ1) is greater than the mean of population 2 (μ2).

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold for determining whether the observed data is statistically significant. In this question, α is set at 0.10, meaning there is a 10% risk of concluding that μ1 is greater than μ2 when it is not.

Independent Samples and Variance

Independent samples refer to two groups that are not related or paired in any way, allowing for the comparison of their means. In this scenario, the assumption that the variances of the two populations are not equal (σ1^2 ≠ σ2^2) indicates the use of a specific statistical test, such as Welch's t-test, which adjusts for the difference in variances when comparing the means.

관련 실천

교과서 질문

In Exercises 25–28, determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions p1 and p2 at the level of significance α. Assume the samples are random and independent.

Claim: p1<p2; α=0.05

Sample statistics: x1 = 86, n1=900 and x2 = 107, n2 = 1200

views

교과서 질문

In Exercises 17 and 18, (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

views

교과서 질문

In Exercises 19–22, test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.

Claim: μd≠0; α=0.05.

Sample statistics: d̄=17.5, sd=4.05, n=37

views

교과서 질문

In Exercises 23 and 24, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) calculate d̄ and sd, (d) find the standardized test statistic t, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.

A physical fitness instructor claims that a weight loss supplement will help users lose weight after two weeks. The table shows the weights (in pounds) of 9 adults before using the supplement and two weeks after using the supplement. At α=0.10, is there enough evidence to support the physical fitness instructor’s claim?

views

교과서 질문

"In Exercises 9 and 10, (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.

A career counselor claims that the mean annual salaries of entry level paralegals in Dayton, Ohio, and Coventry, Rhode Island, are the same. The mean annual salary of 40 randomly selected entry level paralegals in Dayton is \$58,180. Assume the population standard deviation is \$10,990. The mean annual salary of 35 randomly selected entry level paralegals in Coventry is \$61,120. Assume the population standard deviation is \$11,850. At α=0.10, is there enough evidence to reject the counselor’s claim? (Adapted from Salary.com)"

views

교과서 질문

Claim: μd<0; α=0.10.

Sample statistics: d̄=3.2, sd=5.68, n=25

views