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 is H₀: p = 0.25, and the alternative hypothesis is Hₐ: p ≠ 0.25, since this is a two-tailed test.
2. Step 2: Calculate the sample proportion (p̂). The formula for p̂ is 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 a hypothesis test for proportions. Ensure that both np and nq are greater than or equal to 5, where q = 1 - p. For np, calculate np = n * p, and for nq, calculate nq = n * q.
4. Step 4: Compute the test statistic (z-value). The formula for the z-value is 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 α = 0.10 (two-tailed test) or use the p-value approach. If the p-value is less than α, reject the null hypothesis; otherwise, fail to reject the null hypothesis.