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. $n = 82 , p = 0.66 : P ( 44 \leq X \leq 57 )$
A) 0.2148
B) 0.7787
C) 0.0066
D) 0.7852

Question

Quizplus · Accepted Answer

To find $P(44 \leq X \leq 57)$ using the normal approximation, first calculate the mean $\mu = np = 82 \times 0.66 = 54.12$ and the standard deviation $\sigma = \sqrt{np(1-p)} = \sqrt{82 \times 0.66 \times (1-0.66)} \approx 4.57$. Then, convert the X values to Z-scores: $Z_1 = \frac{44 - 54.12}{4.57} \approx -2.21$ and $Z_2 = \frac{57 - 54.12}{4.57} \approx 0.63$. Using Z-tables or a calculator, find $P(Z_1 \leq Z \leq Z_2) = P(-2.21 \leq Z \leq 0.63)$, which is approximately 0.7787.

Use the Normal Approximation to Find the Indicated Probability n=82,p=0.66:P(44≤X≤57)n = 82 , p = 0.66 : P ( 44 \leq X \leq 57 )n=82,p=0.66:P(44≤X≤57)

Use the Normal Approximation to Find the Indicated Probability $n = 82 , p = 0.66 : P ( 44 \leq X \leq 57 )$