A sample of 51 observations will be taken from a process (an infinite population). The population proportion equals 0.85. The probability that the sample proportion will be between 0.9115 and 0.946 is
A)0.0345
B)0.0819
C)0.6900
D)0.8633

Question

Quizplus · Accepted Answer

We are given that the population proportion is p = 0.85 and the sample size is n = 51. Also note that the sample proportion, say p_hat, follows a normal distribution with mean = p = 0.85 and standard deviation = sqrt[p(1-p)/n] = sqrt[(0.85)(0.15)/51] = 0.064.

To find the probability that the sample proportion is between 0.9115 and 0.946, we first standardize the values:
(z_1) = (0.9115 - 0.85)/0.064 = 0.9594
(z_2) = (0.946 - 0.85)/0.064 = 1.5000

Then we look up the area under the standard normal curve between these z-values:
P(0.9594 ≤ Z ≤ 1.5000) = 0.0819

Therefore, the probability that the sample proportion will be between 0.9115 and 0.946 is 0.0819, which corresponds to choice (B).

A Sample of 51 Observations Will Be Taken from a Process