In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.

Claim: ; μ ≠ 5880; α = 0.03; α = 413
Sample statistics: x_bar = 5771, n = 67

검증된 단계별 안내

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: μ = 5880, and the alternative hypothesis is Hₐ: μ ≠ 5880. This is a two-tailed test because the claim specifies 'not equal to' (≠).

Step 2: Calculate the test statistic using the formula for a t-test: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Substitute the given values: x̄ = 5771, μ = 5880, s = 413, and n = 67.

Step 3: Determine the degrees of freedom (df) for the t-distribution. For a single-sample t-test, df = n - 1. In this case, df = 67 - 1 = 66.

Step 4: Find the critical t-value(s) for a two-tailed test at the significance level α = 0.03 and df = 66. Use a t-distribution table or statistical software to find the critical values. These values will define the rejection region for the null hypothesis.

Step 5: Compare the calculated t-value from Step 2 to the critical t-values from Step 4. If the calculated t-value falls in the rejection region (i.e., beyond the critical values), reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of 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 a population 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 null hypothesis would state that the population mean μ equals 5880, while the alternative hypothesis claims that μ does not equal 5880.

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.03, indicating a 3% risk of concluding that a difference exists when there is none. This threshold helps determine the critical value for the test statistic, which is essential for making a decision regarding the null hypothesis.

Sample Mean and Standard Error

The sample mean (x̄) is the average value of a sample, which in this case is 5771. The standard error (SE) measures the dispersion of the sample mean from the population mean and is calculated using the sample standard deviation divided by the square root of the sample size (n). Understanding the sample mean and standard error is crucial for calculating the test statistic and determining whether the sample provides enough evidence to reject the null hypothesis.

In Exercises 29–32, test the claim about the population mean at the level of significance α. Assume the population is normally distributed.Claim: ; μ ≠ 5880; α = 0.03; α = 413Sample statistics: x_bar = 5771, n = 67