In Exercises 13–18, test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed.
Claim: μ≠52,200; α=0.05. Sample statistics: x_bar=53,220, s=2700, n=34

검증된 단계별 안내

Step 1: State the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 52,200, and the alternative hypothesis is H₁: μ ≠ 52,200. This is a two-tailed test because the claim specifies 'not equal to.'

Step 2: Identify the test statistic to use. Since the population standard deviation is not provided and the sample size is relatively small (n = 34), use the t-test. The formula for the t-test statistic is: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean under H₀, s is the sample standard deviation, and n is the sample size.

Step 3: Calculate the degrees of freedom (df) for the t-distribution. The degrees of freedom are given by df = n - 1. In this case, df = 34 - 1 = 33.

Step 4: Determine the critical t-value(s) for a two-tailed test at the significance level α = 0.05. Use a t-distribution table or statistical software to find the critical t-values corresponding to df = 33 and α/2 = 0.025 in each tail.

Step 5: Compare the calculated t-test statistic to the critical t-values. If the absolute value of the t-test statistic exceeds the critical t-value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Additionally, you can calculate the p-value and compare it to α to make the decision.

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주요 개념

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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). In this case, the null hypothesis states that the population mean μ equals 52,200, while the alternative hypothesis claims that it does not. The goal is to determine whether there is enough evidence to reject the null hypothesis at a specified significance level.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.05, indicating a 5% risk of concluding that a difference exists when there is none. This level helps to control the likelihood of false positives in hypothesis testing.

Confidence Interval

A confidence interval is a range of values derived from sample statistics that is likely to contain the population parameter with a certain level of confidence. In the context of hypothesis testing, it can be used to assess whether the population mean falls within a specified range. If the hypothesized mean (52,200) lies outside the confidence interval calculated from the sample data, it provides evidence against the null hypothesis.

In Exercises 13–18, test the claim about the population mean μ at the level of significance α. Assume the population is normally distributed.Claim: μ≠52,200; α=0.05. Sample statistics: x_bar=53,220, s=2700, n=34