1. The test statistic for this sample is -2.828 (accurate to three decimal places).
2. The p-value for this sample is 0.0047 (accurate to four decimal places).
3. The conclusion is that A) There is sufficient evidence to warrant rejection of the claim that the population mean is less than 79.9.
1) Calculating the test statistic:
The test statistic for a one-sample t-test is:
t = (M - μ) / (s / sqrt(n))
Where:
The sample mean = M
The hypothesized population mean = μ
The sample standard deviation = s
The sample size = n
Substituting the given values:
t = (77.4 - 79.9) / (5.8 / sqrt(28)) t = -2.828
Thus, the test statistic for this sample is -2.828.
2) Calculating the p-value:
The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated from the sample, assuming the null hypothesis is true. Since this is a one-tailed test with the alternative hypothesis that the population mean is greater than 79.9, we need to find the area under the t-distribution curve to the right of the test statistic.
Using a t-distribution table or calculator, we find that the area to the right of -2.828 with 27 degrees of freedom is 0.0047.
Thus, the p-value for this sample is 0.0047.
3) We can reject the null hypothesis at a significance level of 0.02 and conclude that the population mean is greater than 79.9.