Questions: 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 n=38 and the test statistic is t=-2.446. Use technology to find the P-value. Based on the result, what is the final conclusion? Use a significance level of 0.01. State the null and alternative hypotheses. H0: μ=5.670 H1: μ ≠ 5.670 The test statistic is -2.45. The P -value is

Solution Steps

To solve this problem, we need to calculate the P-value for the given test statistic using a t-distribution. The null hypothesis states that the mean weight of the quarters is 5.670 grams, and the alternative hypothesis states that it is not equal to 5.670 grams. Given the test statistic and the sample size, we can use Python to find the P-value and then compare it to the significance level to draw a conclusion.

1. Define the null and alternative hypotheses.
2. Use the given test statistic and sample size to find the degrees of freedom.
3. Calculate the P-value using the t-distribution.
4. Compare the P-value to the significance level to determine whether to reject the null hypothesis.

The null and alternative hypotheses are defined as follows: \[ \begin{align_} \mathrm{H}_{0}: & \quad \mu = 5.670 \\ \mathrm{H}_{1}: & \quad \mu \neq 5.670 \end{align_} \]

The degrees of freedom (df) for the t-test is calculated as: \[ \text{df} = n - 1 = 38 - 1 = 37 \]

Using the test statistic \( t = -2.446 \) and the degrees of freedom \( df = 37 \), the P-value is calculated as: \[ \text{P-value} = 0.0193 \]

The significance level is given as \( \alpha = 0.01 \). We compare the P-value with the significance level: \[ 0.0193 > 0.01 \]

Since the P-value is greater than the significance level, we fail to reject the null hypothesis.

Final Answer