Questions: In a memory test, the test subjects are given a large number and are asked to memorize it. Historical records show that 80% of test subjects pass the test. To pass the test, a subject must exactly repeat all the digits in the number after two hours. A random sample of 625 people to take the memory test is going to be chosen. Let p̂ be the proportion of people in the sample who pass the test. Answer the following. (If necessary, consult a list of formulas.) (a) Find the mean of p̂ 0.8 (b) Find the standard deviation of p̂ 0.016 (c) Compute an approximation for P(p̂>0.76), which is the probability that more than 76% of the people in the sample pass the test. Round your answer to four decimal places.

In a memory test, the test subjects are given a large number and are asked to memorize it. Historical records show that 80% of test subjects pass the test. To pass the test, a subject must exactly repeat all the digits in the number after two hours.

A random sample of 625 people to take the memory test is going to be chosen. Let p̂ be the proportion of people in the sample who pass the test.

Answer the following. (If necessary, consult a list of formulas.)
(a) Find the mean of p̂
0.8
(b) Find the standard deviation of p̂
0.016
(c) Compute an approximation for P(p̂>0.76), which is the probability that more than 76% of the people in the sample pass the test. Round your answer to four decimal places.

Transcript text: In a memory test, the test subjects are given a large number and are asked to memorize it. Historical records show that $80 \%$ of test subjects pass the test. To pass the test, a subject must exactly repeat all the digits in the number after two hours. A random sample of 625 people to take the memory test is going to be chosen. Let $\hat{p}$ be the proportion of people in the sample who pass the test. Answer the following. (If necessary, consult a list of formulas.) (a) Find the mean of $\widehat{p}$ 0.8 (b) Find the standard deviation of $\widehat{p}$ 0.016 (c) Compute an approximation for $P(\hat{p}>0.76)$, which is the probability that more than $76 \%$ of the people in the sample pass the test. Round your answer to four decimal places.

Solution

Solution Steps

Step 1: Calculate the mean of $\hat{p}$

The mean of the sample proportion $\hat{p}$ is equal to the population proportion $p$, \[ \mu_{\hat{p}} = p = 0.8 \]

Step 2: Calculate the standard deviation of $\hat{p}$

The standard deviation of the sample proportion $\hat{p}$ can be calculated using the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.8(1-0.8)}{625}}= 0.016 \]

Step 3: Approximate probabilities using the normal distribution

For calculating probabilities like $P(\hat{p} < \hat{p}_0)$ or $P(\hat{p} > \hat{p}_0)$, we use the Z-score formula: \[ Z = \frac{\hat{p}_0 - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.76 - 0.8}{0.016}=-2.5 \] Using standard normal distribution tables or software, we find the corresponding probability for Z = -2.5 is approximately 0.0062.