Question

use a t - test to test the claim about the population mean \\(\mu\\) at the given level of significance \\(\alpha\\) using the given sample statistics. assume the population is normally distributed.\
claim: \\(\mu = 52,600\\); \\(\alpha = 0.01\\) \quad sample statistics: \\(\bar{x} = 51,224\\), \\(s = 2300\\), \\(n = 20\\)\
click the icon to view the t - distribution table.\
\\(h_{a}: \mu = 52,600\\)\
\\(h_{a}: \mu \
eq 52,600\\)\
what is the value of the standardized test statistic?\
the standardized test statistic is \\( - 2.68\\) (round to two decimal places as needed.)\
what is(are) the critical value(s)?

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\). Given \(n = 20\), so \(df=20-1 = 19\).

Step2: Find the critical values for a two - tailed test

Since \(\alpha=0.01\) and it is a two - tailed test, we divide \(\alpha\) by 2. So \(\frac{\alpha}{2}=\frac{0.01}{2}=0.005\).
Looking up the \(t\) - distribution table with \(df = 19\) and the area in the tails \(0.005\) for each tail.
The critical values are \(t_{0}=\pm 2.861\) (from the \(t\) - distribution table).

Answer:

The critical values are \(-2.861\) and \(2.861\).