Question

use technology and a t - test to test the claim about the population mean μ at the given level of significance a given sample statistics. assume the population is normally distributed.
claim: μ>78; α = 0.01 sample statistics: x̄ = 80.1, s = 3.4, n = 25
what are the null and alternative hypotheses? choose the correct answer below.
a. h₀: μ≠78
hₐ: μ = 78
b. h₀: μ≤78
hₐ: μ>78
c. h₀: μ = 78
hₐ: μ≠78
d. h₀: μ≥78
hₐ: μ<78
what is the value of the standardized test statistic?
the standardized test statistic is 3.09. (round to two decimal places as needed.)
what is the p - value of the test statistic?
p - value = (round to three decimal places as needed.)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \( df=n - 1\). Given \(n = 25\), then \(df=25-1 = 24\).

Step2: Use the t - distribution to find the P - value

We have a one - tailed (right - tailed) t - test with \(t = 3.09\) and \(df = 24\). Using a t - distribution table or technology (such as a TI - 84 Plus: tcdf(3.09,1E99,24)), we calculate the P - value.
The formula for the P - value in a right - tailed t - test is \(P(T>t)\) where \(T\sim t_{df}\).

Answer:

The P - value is \(0.003\) (rounded to three decimal places).