Question

use a t - test to test the claim about the population mean μ at the given level of significance α using the given sample statistics. assume the population is normally distributed. claim: μ ≥ 7800; α = 0.05 sample statistics: x̄ = 7500, s = 430, n = 24 what are the null and alternative hypotheses? a. h₀: μ = 7800 hₐ: μ ≠ 7800 b. h₀: μ ≥ 7800 hₐ: μ < 7800 c. h₀: μ ≤ 7800 hₐ: μ > 7800 d. h₀: μ ≠ 7800 hₐ: μ = 7800 what is the value of the standardized test statistic? the standardized test statistic is □. (round to two decimal places as needed.)

Explanation:

Step1: Calculate the standardized test statistic

The formula for the \(t -\)test statistic is \(t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\).
Here, \(\bar{x} = 7500\), \(\mu = 7800\), \(s = 430\), \(n=24\).
Substitute the values into the formula:
\(t=\frac{7500 - 7800}{\frac{430}{\sqrt{24}}}\)
First, calculate \(\sqrt{24}\approx4.899\), then \(\frac{430}{\sqrt{24}}\approx\frac{430}{4.899}\approx87.77\)
Next, \(7500 - 7800=- 300\)
So, \(t=\frac{-300}{87.77}\approx - 3.42\)

Answer:

The standardized test statistic is \(-3.42\)