In Exercises 23 and 24, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) calculate d̄ and sd, (d) find the standardized test statistic t, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.


A physical fitness instructor claims that a weight loss supplement will help users lose weight after two weeks. The table shows the weights (in pounds) of 9 adults before using the supplement and two weeks after using the supplement. At α=0.10, is there enough evidence to support the physical fitness instructor’s claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (Ho) and alternative hypothesis (Ha). The claim is that the weight loss supplement helps users lose weight after two weeks. Ho: μd = 0 (there is no difference in weight before and after using the supplement). Ha: μd < 0 (the weight after using the supplement is less than the weight before).
2. Step 2: Find the critical value(s) and identify the rejection region(s). Since α = 0.10 and the test is one-tailed (left-tailed), use a t-distribution table with degrees of freedom (df = n - 1 = 9 - 1 = 8) to find the critical value. The rejection region is t < critical value.
3. Step 3: Calculate the mean difference (d̄) and the standard deviation of the differences (sd). Compute the differences for each user (Weight before - Weight after), then calculate d̄ = Σd / n and sd = sqrt[Σ(d - d̄)^2 / (n - 1)].
4. Step 4: Find the standardized test statistic t using the formula t = (d̄ - 0) / (sd / sqrt(n)), where n is the number of paired observations.
5. Step 5: Decide whether to reject or fail to reject the null hypothesis. Compare the calculated t-value with the critical value. If t < critical value, reject Ho; otherwise, fail to reject Ho. Interpret the decision in the context of the original claim: determine whether there is enough evidence to support the instructor's claim that the supplement helps users lose weight.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. In this context, the instructor's claim about the weight loss supplement serves as the alternative hypothesis.

Dependent Samples

Dependent samples refer to pairs of observations that are related or matched in some way, often used in before-and-after studies. In this case, the weights of the same individuals before and after using the supplement are compared, making the samples dependent. This relationship necessitates the use of specific statistical tests, such as the paired t-test, to analyze the differences in weights.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which in this case is set at 0.10. The rejection region is the range of values for the test statistic that would lead to rejecting H0. Understanding these concepts is crucial for interpreting the results of the statistical test and making informed decisions based on the data.