Skip to main content
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.RE.8
8장, 문제 8.RE.8

In Exercises 5–8, 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=0.05


Population statistics: σ1= 14 and σ2= 15


Sample statistics: x̅1 = 87, n1 = 410, and x̅2= 85, n2= 340

검증된 단계별 안내
1
Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\) based on the claim. Since the claim is \(\mu_1 \neq \mu_2\), set \(H_0: \mu_1 = \mu_2\) and \(H_a: \mu_1 \neq \mu_2\).
Determine the significance level \(\alpha = 0.05\) and note that this is a two-tailed test because the alternative hypothesis is not equal to (\(\neq\)).
Calculate the standard error of the difference between the two sample means using the formula: \(SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\) where \(\sigma_1 = 14\), \(\sigma_2 = 15\), \(n_1 = 410\), and \(n_2 = 340\).
Compute the test statistic (z-score) using the formula: \(z = \frac{\bar{x}_1 - \bar{x}_2}{SE}\) where \(\bar{x}_1 = 87\) and \(\bar{x}_2 = 85\) are the sample means.
Find the critical z-values for a two-tailed test at \(\alpha = 0.05\), which correspond to the z-scores that cut off the upper and lower 2.5% of the standard normal distribution. Compare the calculated test statistic to these critical values to decide whether to reject or fail to reject the null hypothesis.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
5m
도움이 되었나요?

주요 개념

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

Hypothesis Testing for Two Population Means

This involves testing whether there is a statistically significant difference between two population means (μ1 and μ2). The null hypothesis typically states that the means are equal (μ1 = μ2), while the alternative reflects the claim (μ1 ≠ μ2). The test uses sample data to decide whether to reject the null hypothesis at a given significance level (α).
추천 영상:
가이드 코스
08:24
Difference in Means: Hypothesis Tests

Level of Significance (α)

The level of significance, α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). Commonly set at 0.05, it defines the threshold for deciding if the observed difference is statistically significant. If the p-value is less than α, the null hypothesis is rejected.
추천 영상:
가이드 코스
03:33
Finding Binomial Probabilities Using TI-84 Example 1

Sampling Distribution and Standard Error

The sampling distribution of the difference between sample means is approximately normal when samples are large or populations are normal. The standard error measures the variability of this difference and is calculated using population standard deviations and sample sizes. It is essential for computing the test statistic in hypothesis testing.
추천 영상:
가이드 코스
05:11
Sampling Distribution of Sample Proportion
관련 실천
교과서 질문

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=0.01. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1= 44.5, s1= 5.85, n1= 17 and x̅2= 49.1, s2= 5.25, n2= 18

55
views
교과서 질문

In Exercises 17 and 18, (c) find the standardized test statistic t, 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.

38
views
교과서 질문

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The fuel efficiencies of 12 cars

Sample 2: The fuel efficiencies of the same 12 cars using an alternative fuel




60
views
교과서 질문

"In Exercises 17 and 18, (b) find the critical value(s) and identify the rejection region(s), 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."

36
views
교과서 질문

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.


Sample 1: The weights of 45 oranges

Sample 2: The weights of 40 grapefruits


71
views
교과서 질문

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=0.01. Assume (σ1)^2 = (σ2)^2


Sample statistics: x̅1= 61, s1= 3.3, n1= 5 and x̅2= 55.1, s2= 1.2, n2= 7

50
views