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=0.10.


Sample statistics: d̄=3.2, sd=5.68, n=25

Verified step by step guidance

1. Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that μd = 0, while the alternative hypothesis (H₁) states that μd < 0. This is a one-tailed test since the claim is μd < 0.
2. Step 2: Calculate the test statistic using the formula for the t-test for paired data: t = (d̄ - μd) / (sd / √n). Here, d̄ is the sample mean of the differences, sd is the standard deviation of the differences, n is the sample size, and μd is the hypothesized mean difference (0 under H₀).
3. Step 3: Determine the degrees of freedom (df) for the t-test. For paired data, df = n - 1. In this case, df = 25 - 1 = 24.
4. Step 4: Find the critical t-value for a one-tailed test at the significance level α = 0.10 with df = 24. Use a t-distribution table or statistical software to find this value.
5. Step 5: Compare the calculated t-statistic to the critical t-value. If the t-statistic is less than the critical t-value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis.

Key Concepts

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 less than zero, which will be tested against the significance level α.

Level of Significance (α)

The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. In this scenario, α is set at 0.10, indicating a 10% risk of concluding that a difference exists when there is none.