Questions: (a) Write the claim mathematically and identify H₀ and Hₐ. Which of the following correctly states H₀ and Hₐ ? A. H₀: μ=32 B. H₀: μ<32 C. H₀: μ>32 Therefore Hₐ: μ ≠ 32 Hₐ: μ ≥ 32 Hₐ: μ ≤ 32 D. H₀: μ ≥ 32 E. H₀: μ=32 F. H₀: μ ≤ 32 Hₐ: μ<32 Hₐ: μ<32 Hₐ: μ>32 (b) Use technology to find the P-value. P= (Round to three decimal places as needed.)

(a) Write the claim mathematically and identify H₀ and Hₐ.

Which of the following correctly states H₀ and Hₐ ?
A. H₀: μ=32
B. H₀: μ<32
C. H₀: μ>32
Therefore Hₐ: μ ≠ 32 Hₐ: μ ≥ 32
Hₐ: μ ≤ 32
D. H₀: μ ≥ 32
E. H₀: μ=32
F. H₀: μ ≤ 32 Hₐ: μ<32 Hₐ: μ<32 Hₐ: μ>32
(b) Use technology to find the P-value.
P= (Round to three decimal places as needed.)

Transcript text: (a) Write the claim mathematically and identify $\mathrm{H}_{0}$ and $\mathrm{H}_{a}$. Which of the following correctly states $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$ ? A. $\mathrm{H}_{0}: \mu=32$ B. $H_{0}: \mu<32$ C. $\mathrm{H}_{0}: \mu>32$ $\therefore H_{a}: \mu \neq 32$ $H_{a}: \mu \geq 32$ $\mathrm{H}_{\mathrm{a}}: \mu \leq 32$ D. $H_{0}: \mu \geq 32$ E. $H_{0}: \mu=32$ F. $\mathrm{H}_{0}: \mu \leq 32$ $H_{a}: \mu<32$ $H_{a}: \mu<32$ $H_{a}: \mu>32$ (b) Use technology to find the P-value. $P=$ $\square$ (Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Formulate Hypotheses

Null Hypothesis ($H_0$): $\mu \geq 32$ Alternative Hypothesis ($H_a$): $\mu < 32$

Step 2: Calculate the Test Statistic

Using the T-test formula: $T = \frac{\bar{x} - \mu}{s/\sqrt{n}} = -1.031$

Step 3: Find the P-value

The p-value is 0.16, which is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.

Step 4: Make a Decision

Since $p = 0.16 > \alpha = 0.05$, we do not reject the null hypothesis.

Step 5: Interpret the Decision

Based on the p-value of 0.16 and our level of significance of 0.05, we do not have sufficient evidence to reject the null hypothesis. Therefore, we cannot support the claim that the population mean is less than 32.