[APPLET] A weight loss program claims that program participants have a mean weight loss of at least 10.5 pounds after 1 month. The weight losses after 1 month (in pounds) of a random sample of 40 program participants are listed below. At α=0.01, is there enough evidence to reject the program’s claim?

검증된 단계별 안내

Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) is that the mean weight loss is at least 10.5 pounds (H₀: μ ≥ 10.5). The alternative hypothesis (H₁) is that the mean weight loss is less than 10.5 pounds (H₁: μ < 10.5).

Step 2: Calculate the sample mean (x̄) and sample standard deviation (s) using the provided data. Use the formulas for the sample mean (x̄ = Σx / n) and the sample standard deviation (s = √[Σ(x - x̄)² / (n - 1)]), where n is the sample size.

Step 3: Determine the test statistic. Since the population standard deviation is not provided, use the t-test formula: t = (x̄ - μ) / (s / √n), where μ is the hypothesized population mean (10.5 pounds), x̄ is the sample mean, s is the sample standard deviation, and n is the sample size.

Step 4: Find the critical value for the t-test at a significance level of α = 0.01 with degrees of freedom (df = n - 1). Use a t-distribution table or statistical software to find the critical t-value for a one-tailed test.

Step 5: Compare the calculated t-test statistic to the critical t-value. If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem.

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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 would state that the mean weight loss is less than or equal to 10.5 pounds, while the alternative hypothesis would assert that it is greater than 10.5 pounds. The goal is to determine if there is enough evidence to reject the null hypothesis at a specified significance level (α).

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 0.01, indicating a 1% risk of concluding that the program's claim is false when it is actually true. A lower α value means a stricter criterion for evidence against the null hypothesis.

Sample Mean and Standard Deviation

The sample mean is the average weight loss of the participants in the study, calculated by summing all the individual weight losses and dividing by the number of participants. The standard deviation measures the variability of the weight losses around the mean. These statistics are crucial for conducting hypothesis tests, as they help determine the test statistic, which is used to compare against critical values to decide whether to reject the null hypothesis.