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

검증된 단계별 안내

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) is that the mean of the differences μd = 0, and the alternative hypothesis (H₁) is that μd ≠ 0. This is a two-tailed test since the claim is μd ≠ 0.

Step 2: Calculate the test statistic using the formula t = (d̄ - μd) / (sd / √n), where d̄ is the sample mean of the differences, μd is the hypothesized population mean of the differences (0 in this case), sd is the sample standard deviation of the differences, and n is the sample size.

Step 3: Determine the degrees of freedom (df) for the t-distribution. The degrees of freedom are calculated as df = n - 1, where n is the sample size.

Step 4: Find the critical t-value(s) for a two-tailed test at the significance level α = 0.05 and the calculated degrees of freedom. Use a t-distribution table or statistical software to find the critical values.

Step 5: Compare the calculated test statistic to the critical t-values. If the test statistic falls outside the range defined by the critical t-values, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

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주요 개념

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Paired Data

Paired data refers to two sets of related observations, often collected from the same subjects under different conditions. This type of data is used in statistical tests to determine if there is a significant difference between the two conditions. In this context, the differences between paired observations are analyzed to test claims about their mean.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences 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 this case, the claim is that the mean of the differences (μd) is not equal to zero, which is tested against a significance level (α) of 0.05.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether a statistical result is significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, an α of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none, guiding the decision-making process in hypothesis testing.

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