Use the normal approximation to find the indicated probability. The sample size is $n$, the population proportion of successes is $p$, and $X$ is the number of successes in the sample.
$$n = 75 , p = 0.52 : P ( 34 \leq X \leq 47 )$$
A) 0.0250
B) 0.1020
C) 0.9750
D) 0.8730

Question

Quizplus · Accepted Answer

To find $P(34 \leq X \leq 47)$ using the normal approximation, we first calculate the mean ($\mu$) and standard deviation ($\sigma$) of the binomial distribution. The formulas for these are $\mu = np$ and $\sigma = \sqrt{np(1-p)}$, respectively.Given $n = 75$ and $p = 0.52$, we find:- $\mu = 75 \times 0.52 = 39$- $\sigma = \sqrt{75 \times 0.52 \times (1 - 0.52)} = \sqrt{75 \times 0.52 \times 0.48} \approx 4.34$To use the normal approximation, we apply the continuity correction because we're dealing with a discrete distribution (binomial) and approximating it with a continuous distribution (normal). This means we adjust our bounds slightly to $P(33.5 \leq X \leq 47.5)$.Next, we convert these bounds to z-scores:- $z_1 = \frac{33.5 - 39}{4.34} \approx -1.27$- $z_2 = \frac{47.5 - 39}{4.34} \approx 1.96$Using standard normal distribution tables or a calculator, we find:- $P(Z \leq -1.27) \approx 0.1020$- $P(Z \leq 1.96) \approx 0.9750$The probability we're interested in is the area between these z-scores, so we subtract:$$P(34 \leq X \leq 47) = P(Z \leq 1.96) - P(Z \leq -1.27) \approx 0.9750 - 0.1020 = 0.8730.$$Thus, the correct answer is D, 0.8730.

Use the Normal Approximation to Find the Indicated Probability nnn , the Population Proportion of Successes Is

Use the Normal Approximation to Find the Indicated Probability $n$ , the Population Proportion of Successes Is