A test of sobriety involves measuring the subject's motor skills. A sample of 33 randomly selected sober subjects take the test and produce a mean score of 68.1 with a standard deviation of 4 . A claim is made that the true mean score for all sober subjects is not equal to 67.

For each part below, enter only a numeric value in the answer box. For example, do not type " $z=$ " or " $t=$ " before your answers. Round each of your answers to 3 places after the decimal point.
(a) Calculate the value of the test statistic used in this test.
Test statistic's value $=$
(b) Use your calculator to find the $P$-value of this test.
\[
P \text {-value }=
\]
(c) Use your calculator to find the critical value(s) used to test this claim at the 0.05 significance level. If there are two critical values, then list them both with a comma between them.
Critical value $(\mathrm{s})=$

Step 1 ：Given that the sample size is 33, the sample mean is 68.1, the sample standard deviation is 4, and the population mean is 67.

Step 2 ：Calculate the test statistic using the formula for a one-sample t-test, which is \((\text{sample mean} - \text{population mean}) / (\text{sample standard deviation} / \sqrt{\text{sample size}})\). The calculated test statistic is approximately 1.580.

Step 3 ：Calculate the p-value using a t-distribution table or a statistical calculator. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The calculated p-value is approximately 0.124.

Step 4 ：Find the critical value(s) using a t-distribution table or a statistical calculator. The critical value is the value of the test statistic that separates the region where we would reject the null hypothesis from the region where we would not reject the null hypothesis. Since this is a two-tailed test (because the alternative hypothesis is that the true mean is not equal to 67), there will be two critical values. The calculated critical values are approximately -2.037 and 2.037.

Step 5 ：\(\boxed{\text{The test statistic's value is approximately 1.580.}}\)

Step 6 ：\(\boxed{\text{The p-value of this test is approximately 0.124.}}\)

Step 7 ：\(\boxed{\text{The critical values used to test this claim at the 0.05 significance level are approximately -2.037 and 2.037.}}\)