Finding P-values
In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?


Weights of Quarters The claim is that weights (grams) of quarters made after 1964 have a mean equal to 5.670 g as required by mint specifications. The sample size is and the test statistic is t = -3.135.

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 5.670 g, which states that the mean weight of quarters is equal to 5.670 g. The alternative hypothesis is H₁: μ ≠ 5.670 g, which indicates a two-tailed test.
2. Step 2: Determine the degrees of freedom (df) for the t-distribution. The degrees of freedom are calculated as df = n - 1, where n is the sample size. Ensure you know the sample size to compute df.
3. Step 3: Use the given test statistic t = -3.135 and the degrees of freedom to find the P-value. You can either use statistical software or a t-distribution table (Table A-3). For a two-tailed test, double the area in the tail corresponding to |t| = 3.135.
4. Step 4: Compare the P-value to the significance level (α). If the P-value is less than α (commonly 0.05), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Based on the comparison, draw a conclusion. If the null hypothesis is rejected, conclude that there is sufficient evidence to support the claim that the mean weight of quarters is not equal to 5.670 g. If the null hypothesis is not rejected, conclude that there is insufficient evidence to refute the claim.

Key Concepts

P-value

The P-value is a statistical measure that helps determine the significance of results from a hypothesis test. It represents the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, often leading to its rejection if it falls below a predetermined significance level, such as 0.05.

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 a default position, and the alternative hypothesis (H1), which represents what we aim to support. The test statistic, such as the t-value in this case, is calculated to assess the evidence against the null hypothesis.

Test Statistic

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It quantifies the difference between the observed sample statistic and the hypothesized population parameter, scaled by the standard error. In this scenario, the t-statistic of -3.135 indicates how many standard errors the sample mean is away from the hypothesized mean, providing a basis for determining the P-value.