Question

if a sample proportion is 0.70, which of the following could be a 90% confidence interval for the population proportion? select all that apply a. lower bound 0.68, upper bound 0.74 b. lower bound 0.52, upper bound 0.79 c. lower bound 0.61, upper bound 0.79 d. lower bound 0.79, upper bound 0.97 e. lower bound 0.67, upper bound 0.73

Explanation:

Step1: Calculate the margin of error for each option

The sample proportion \(p = 0.70\). For a confidence interval \((a,b)\), the margin of error \(E=\frac{b - a}{2}\), and the center of the confidence interval is \(\frac{a + b}{2}\). Since the center of the confidence interval for proportion should be the sample proportion \(p\) (because \(\text{Center}=\frac{\text{Lower bound}+\text{Upper bound}}{2}\)), we check for each option:

• Option A: \(\text{Center}=\frac{0.68 + 0.74}{2}=\frac{1.42}{2}=0.71

eq0.70\)

• Option B: \(\text{Center}=\frac{0.52+0.79}{2}=\frac{1.31}{2}=0.655

eq0.70\)

• Option C: \(\text{Center}=\frac{0.61 + 0.79}{2}=\frac{1.4}{2}=0.70\), \(E=\frac{0.79 - 0.61}{2}=\frac{0.18}{2}=0.09\). For a \(90\%\) confidence interval, the critical value \(z_{\alpha/2}=1.645\), and the formula for the margin of error for proportion \(E = z_{\alpha/2}\sqrt{\frac{p(1 - p)}{n}}\). Although we don't know \(n\), the center is correct.
• Option D: \(\text{Center}=\frac{0.79+0.97}{2}=\frac{1.76}{2}=0.88

eq0.70\)

• Option E: \(\text{Center}=\frac{0.67+0.73}{2}=\frac{1.4}{2}=0.70\), \(E=\frac{0.73 - 0.67}{2}=\frac{0.06}{2}=0.03\). Although we don't know \(n\), the center is correct.

Answer:

C. Lower bound \(0.61\), Upper bound \(0.79\); E. Lower bound \(0.67\), Upper bound \(0.73\)