Questions: State your conclusion to the hypothesis test. If a 95% confidence interval for μ is calculated to be (21.4,27.9), what would be the result of a H0: μ=27 vs H1: μ ≠ 27 The null hypothesis would not be rejected

State your conclusion to the hypothesis test.
If a 95% confidence interval for μ is calculated to be (21.4,27.9), what would be the result of a
H0: μ=27 vs H1: μ ≠ 27

The null hypothesis would not be rejected

Transcript text: State your conclusion to the hypothesis test. If a 95\% confidence interval for $\mu$ is calculated to be $(21.4,27.9)$, what would be the result of a \[ H_{0}: \mu=27 \quad \text { vs } \quad H_{1}: \mu \neq 27 \] The null hypothesis would not be rejected

Solution

Solution Steps

Step 1: Hypothesis Setup

We are testing the following hypotheses:

Null Hypothesis: $ H_0: \mu = 27 $
Alternative Hypothesis: $ H_1: \mu \neq 27 $

Step 2: Calculate Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3.0}{\sqrt{30}} \approx 0.5477 \]

Step 3: Calculate Test Statistic

The test statistic $ Z_{\text{test}} $ is calculated as follows: \[ Z_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{24.65 - 27}{0.5477} \approx -4.2905 \]

Step 4: Calculate P-value

For a two-tailed test, the p-value is calculated using: \[ P = 2 \times (1 - T(|z|)) \approx 0.0 \]

Step 5: Conclusion

Since the p-value $ 0.0 $ is less than the significance level $ \alpha = 0.05 $, we reject the null hypothesis.