Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:
$\alpha =0.01$, $n=40$, $x=28$, & claim is $p=0.75$

• A. Because $P$-value = 0.465 > $\alpha=$ 0.01, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75
• B. Because $P$-value = 0.465 > $\alpha=$ 0.01, we REJECT $H_0$. There is ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75
• C. Because $P$-value = 0.233 > $\alpha=$ 0.01, we REJECT $H_0$. There is ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75
• D. Because $P$-value = 0.233 > $\alpha=$ 0.01, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75

Answer: (A)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: p = 0.75, and the alternative hypothesis is Hₐ: p ≠ 0.75, since this is a two-tailed test.
2. Step 2: Calculate the sample proportion (p̂). The formula for the sample proportion is p̂ = x / n, where x is the number of successes (28) and n is the sample size (40).
3. Step 3: Verify the conditions for using the normal approximation. Check if np ≥ 5 and nq ≥ 5, where q = 1 - p. Substitute the values of n, p, and q to ensure these conditions are met.
4. Step 4: Compute the test statistic (z). The formula for the z-value is z = (p̂ - p) / √[p(1 - p) / n]. Substitute the values of p̂, p, n, and q into the formula.
5. Step 5: Compare the calculated z-value to the critical z-value for α = 0.01 in a two-tailed test. Alternatively, calculate the p-value and compare it to α. If the p-value > α, fail to reject H₀; otherwise, reject H₀.