Technology
In Exercises 9–12, test the given claim by using the display provided from technology. Use a 0.05 significance level. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Peanut Butter Cups Data Set 38 “Candies” includes weights of Reese’s peanut butter cups. The accompanying Statdisk display results from using all 38 weights to test the claim that the sample is from a population with a mean equal to 8.953 g.

검증된 단계별 안내

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that the population mean is equal to 8.953 g (H₀: μ = 8.953). The alternative hypothesis (H₁) states that the population mean is not equal to 8.953 g (H₁: μ ≠ 8.953). This is a two-tailed test.

Step 2: Determine the test statistic. From the Statdisk display, the test statistic is t = -3.42304. This value measures how far the sample mean deviates from the hypothesized population mean in terms of standard errors.

Step 3: Identify the critical t-value. The critical t-value for a two-tailed test at a significance level of 0.05 is ±2.02619, as shown in the Statdisk display. This value defines the rejection regions for the null hypothesis.

Step 4: Compare the test statistic to the critical t-value. Since the test statistic t = -3.42304 falls outside the range defined by the critical t-values (±2.02619), it lies in the rejection region.

Step 5: Evaluate the P-value. The P-value is 0.00153, which is less than the significance level of 0.05. This indicates strong evidence against the null hypothesis. Based on the test statistic and P-value, the null hypothesis is rejected, and the conclusion is that the sample is not from a population with a mean equal to 8.953 g.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents a statement of no effect or no difference, while the alternative hypothesis (H1) suggests that there is an effect or a difference. For this question, the null hypothesis would state that the mean weight of the peanut butter cups is equal to 8.953 g, while the alternative hypothesis would claim that it is not equal to this value.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It measures how far the sample mean is from the null hypothesis mean, in terms of standard errors. In this case, the test statistic is -3.42304, indicating that the sample mean is significantly lower than the hypothesized mean of 8.953 g.

P-Value

The P-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A low P-value (typically less than the significance level, such as 0.05) indicates strong evidence against the null hypothesis. Here, the P-value of 0.00153 suggests that there is strong evidence to reject the null hypothesis, supporting the claim that the mean weight is not equal to 8.953 g.