Use the Central Limit Theorem to find the indicated probability. The sample size is n, the population proportion is p, and the sample proportion is $\hat { p }$ $n = 126 , p = 0.4 ; P ( 0.35

Question

Use the Central Limit Theorem to find the indicated probability. The sample size is n, the population proportion is p, and the sample proportion is $\hat { p }$ $n = 126 , p = 0.4 ; P ( 0.35 < \hat { p } < 0.42 )$
A) 0.1251
B) 0.3228
C) 0.5522
D) 0.6772

Quizplus · Accepted Answer

According to the Central Limit Theorem, when the sample size is sufficiently large (n > 30), the distribution of sample proportions is approximately normal with mean equal to the population proportion (p) and standard deviation equal to the square root of [(p * (1-p))/n].

In this case, n = 126 and p = 0.4, so the mean of the sample proportion is 0.4 and the standard deviation is sqrt[(0.4 * 0.6)/126] = 0.0509.

To find P(0.35 < p-hat < 0.42), we need to standardize the values using the formula z = (x - mu) / sigma, where x is the value in question, mu is the mean, and sigma is the standard deviation.

P(0.35 < p-hat < 0.42) can be rewritten as P((0.35 - 0.4) / 0.0509 < z < (0.42 - 0.4) / 0.0509) = P(-0.98 < z < 0.39).

Using a standard normal table or calculator, we can find that P(-0.98 < z < 0.39) is approximately 0.5522.

Therefore, the answer is C.

Use the Central Limit Theorem to Find the Indicated Probability p^\hat { p }p^​

Use the Central Limit Theorem to Find the Indicated Probability $\hat { p }$