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=0.10, Sample statistics: d̄ =6.5, sd=9.54, n=16

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The claim is μd ≤ 0, so the null hypothesis is H₀: μd ≤ 0, and the alternative hypothesis is H₁: μd > 0 (this is a right-tailed test).

Step 2: Calculate the test statistic using the formula for a t-test for paired data: t = (d̄ - μ₀) / (sd / √n), where d̄ is the sample mean of the differences, μ₀ is the hypothesized mean difference (0 in this case), sd is the standard deviation of the differences, and n is the sample size.

Step 3: Determine the degrees of freedom (df) for the t-distribution. For paired data, df = n - 1. In this case, df = 16 - 1 = 15.

Step 4: Find the critical value for the t-distribution at the given significance level α = 0.10 for a right-tailed test with df = 15. Use a t-table or statistical software to find this value.

Step 5: Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Paired Data

Paired data refers to two sets of related observations, often collected from the same subjects under different conditions. This dependency allows for the analysis of differences between paired observations, which is crucial for tests like the paired t-test. Understanding paired data is essential for accurately interpreting results and making valid inferences about the population from which the samples are drawn.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this context, the null hypothesis states that the mean of the differences (μd) is less than or equal to zero. The level of significance (α) indicates the probability of making a Type I error, which is rejecting a true null hypothesis, and is set at 0.10 in this case.

t-Distribution

The t-distribution is a probability distribution used in hypothesis testing when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the increased variability in smaller samples. In this scenario, the t-distribution is used to calculate the test statistic based on the sample mean difference, standard deviation, and sample size, allowing for inference about the population mean difference.

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