Question

a sample mean, sample size, and sample standard deviation are provided below. use the one - mean t - test to perform the required hypothesis test at the 5% significance level.
\\( \bar{x}=26, n = 7, s = 32, h_{0}: \mu = 28, h_{a}: \mu<28 \\)
click here to view a partial table of values of \\( t_{a} \\).
the test statistic is \\( t= \\)
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the one - mean t - test statistic

The formula for the one - mean t - test statistic is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the hypothesized population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step2: Substitute the given values into the formula

Given \(\bar{x} = 26\), \(\mu=28\), \(s = 7\), and \(n = 32\).
First, calculate \(s/\sqrt{n}\):
\(s/\sqrt{n}=\frac{7}{\sqrt{32}}\approx\frac{7}{5.656854}\approx1.24\)
Then, calculate \(t\):
\(t=\frac{26 - 28}{7/\sqrt{32}}=\frac{- 2}{7/\sqrt{32}}\approx\frac{-2}{1.24}\approx - 1.61\)

Answer:

\(-1.61\)