Question

test the claim about the population mean μ at the level of significance α. assume the population is normally distributed

claim μ > 1473, α = 0.09, σ = 28. sample statistics: x̄ = 1464, n = 28

h₀: μ ≤ 1473 hₐ: μ < 1473

calculate the standardized test statistic
the standardized test statistic is -1.70
(round to two decimal places as needed.)
determine the p - value
p = 0.0445 (round to three decimal places as needed.)
determine the outcome and conclusion of the test
fail to reject h₀. at the 9% significance level, there is enough evidence to support the claim

Explanation:

Step1: Identify Test Type

This is a one - tailed z - test (since population standard deviation \(\sigma\) is known and population is normal) for the population mean. The claim is \(\mu< 1473\), so the alternative hypothesis \(H_{a}:\mu < 1473\) and the null hypothesis \(H_{0}:\mu\geq1473\) (or \(H_{0}:\mu = 1473\) for testing purposes).

Step2: Recall Z - Test Statistic Formula

The formula for the z - test statistic when testing a claim about the population mean (known \(\sigma\)) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\)

Given \(\bar{x} = 1464\), \(\mu = 1473\), \(\sigma=28\), \(n = 28\)

Substitute the values:
\(z=\frac{1464 - 1473}{\frac{28}{\sqrt{28}}}=\frac{- 9}{\frac{28}{5.2915}}\approx\frac{-9}{5.2915}\approx - 1.70\) (matches the given test statistic)

Step3: Find P - value

For a left - tailed test (\(H_{a}:\mu < 1473\)) and \(z=- 1.70\), the P - value is \(P(Z < - 1.70)\)

Using the standard normal table, \(P(Z < - 1.70)=0.0446\approx0.045\) (when rounded to three decimal places)

Step4: Compare P - value and Significance Level

The significance level \(\alpha = 0.09\)

Since \(P - value=0.0446<\alpha = 0.09\), we reject the null hypothesis? Wait, wait, there is a mistake in the original conclusion. Wait, let's re - check.

Wait, the original problem's conclusion says "Fail to reject \(H_{0}\)", but let's recalculate the P - value correctly.

Wait, the z - statistic is \(z=- 1.70\), for a left - tailed test, \(P - value = P(Z < - 1.70)\)

From standard normal table, \(P(Z < - 1.70)=0.0446\)

\(\alpha=0.09\), since \(0.0446<0.09\), we should reject \(H_{0}\)

Wait, maybe there was a mis - identification of the null and alternative. Wait, the claim is \(\mu < 1473\), so \(H_{a}:\mu < 1473\), \(H_{0}:\mu\geq1473\)

The decision rule: Reject \(H_{0}\) if \(P - value<\alpha\)

Since \(P - value = 0.0446<0.09=\alpha\), we reject \(H_{0}\). At the 9% significance level, there is enough evidence to support the claim that \(\mu < 1473\)

But the original problem's conclusion has an error. Let's correct the steps:

Step1: Correct Hypothesis Setup

Claim: \(\mu < 1473\), so \(H_{a}:\mu < 1473\), \(H_{0}:\mu\geq1473\)

Step2: Test Statistic Calculation (Re - check)

\(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{1464 - 1473}{\frac{28}{\sqrt{28}}}\)

\(\sqrt{28}\approx5.2915\), \(\frac{28}{5.2915}\approx5.29\)

\(1464 - 1473=-9\), \(z=\frac{-9}{5.29}\approx - 1.70\) (correct)

Step3: P - value Calculation

For \(z = - 1.70\) and left - tailed test, \(P - value=P(Z < - 1.70)=0.0446\approx0.045\)

Step4: Decision Making

Since \(P - value = 0.0446<\alpha=0.09\), we reject \(H_{0}\). At the 9% significance level, there is enough evidence to support the claim that \(\mu < 1473\)

Answer:

The correct decision is: Reject \(H_{0}\). At the 9% significance level, there is enough evidence to support the claim that \(\mu < 1473\) (The original conclusion in the problem has an error. The correct P - value comparison shows that we should reject \(H_{0}\) because \(0.0446<0.09\))