Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:
$\alpha =0.10$, $n=100$, $x=42$, & claim is $p=0.25$

• A. Because $P$-value = 0.00004 < $\alpha=$ 0.01, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75
• B. Because $P$-value = 0.00004 < $\alpha=$ 0.01, we REJECT $H_0$. There is ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75
• C. Because $P$-value = 0.00008 < $\alpha=$ 0.01, we REJECT $H_0$. There is ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75
• D. Because $P$-value = 0.00008 < $\alpha=$ 0.01, we FAIL TO REJECT $H_0$. There is NOT ENOUGH evidence to suggest $H_{a}$: $p$ ≠ 0.75

Answer: (C)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis typically states that the population proportion is equal to the claimed value (p = 0.25). The alternative hypothesis for a two-tailed test states that the population proportion is not equal to the claimed value (p ≠ 0.25).
2. Step 2: Calculate the sample proportion (p̂). The sample proportion is given by the formula p̂ = x / n, where x is the number of successes (42) and n is the sample size (100).
3. Step 3: Verify the conditions for performing the hypothesis test. Ensure that the sample size is large enough by checking that np ≥ 5 and nq ≥ 5, where q = 1 - p. For this problem, calculate np = n * p and nq = n * q.
4. Step 4: Compute the test statistic (z). Use the formula z = (p̂ - p) / √(p * q / n), where p̂ is the sample proportion, p is the claimed proportion, q = 1 - p, and n is the sample size.
5. Step 5: Compare the calculated z-value to the critical z-value for a two-tailed test at α = 0.10. Alternatively, calculate the p-value and compare it to α. If the p-value is less than α, reject the null hypothesis; otherwise, fail to reject the null hypothesis.