https://file.aiquickdraw.com/mathbot/file/1729274441626llglv8td.jpg

Let's go through the problem step by step:

### Given Information:
- Historical records show that 80% of test subjects pass the test, so \( p = 0.80 \).
- Sample size: \( n = 625 \).

Let’s answer the questions:

---

### (a) Find the mean of \( \hat{p} \).

The mean of \( \hat{p} \) (which is the sample proportion) in a binomial distribution is given by:

\[
\mu_{\hat{p}} = p
\]

Where \( p \) is the population proportion.

\[
\mu_{\hat{p}} = 0.80
\]

So, the mean of \( \hat{p} \) is **0.80**.

---

### (b) Find the standard deviation of \( \hat{p} \).

The standard deviation of \( \hat{p} \) is given by the formula:

\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}
\]

Substitute the values \( p = 0.80 \), \( 1 - p = 0.20 \), and \( n = 625 \):

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.80 \times 0.20}{625}} = \sqrt{\frac{0.16}{625}} = \sqrt{0.000256} = 0.016
\]

So, the standard deviation of \( \hat{p} \) is **0.016**.

---

### (c) Compute an approximation for \( P(\hat{p} > 0.76) \).

We are asked to find the probability that the proportion of people passing the test is greater than 76%, or \( P(\hat{p} > 0.76) \).

To compute this, we need to standardize \( \hat{p} = 0.76 \) using a z-score:

\[
z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}
\]

Substitute the values \( \hat{p} = 0.76 \), \( \mu_{\hat{p}} = 0.80 \), and \( \sigma_{\hat{p}} = 0.016 \):

\[
z = \frac{0.76 - 0.80}{0.016} = \frac{-0.04}{0.016} = -2.5
\]

Now, use the z-score table or normal distribution calculator to find the probability corresponding to \( z = -2.5 \).

From standard z-tables, \( P(z > -2.5) \) is approximately:

\[
P(z > -2.5) = 0.9938
\]

So, the probability that more than 76% of the people pass the test is approximately **0.9938**.

---

### Summary of Answers:
- (a) Mean of \( \hat{p} \): **0.80**.
- (b) Standard deviation of \( \hat{p} \): **0.016**.
- (c) Approximation for \( P(\hat{p} > 0.76) \): **0.9938**.

Would you like more details or explanations on any of these steps?

---

Here are some relative questions to expand on this:

1. How does the sample size affect the standard deviation of the sample proportion?
2. Can the mean \( \hat{p} \) change if the population proportion remains the same?
3. How would the calculation differ if the sample size was smaller or larger?
4. What is the probability that less than 78% of the sample passes the test?
5. How can we compute confidence intervals for \( \hat{p} \)?

**Tip:** Understanding how to compute and use the z-score will help you analyze probabilities for normal distributions effectively.

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. Find the mean of p̂, the standard deviation of p̂, and an approximation for P(p̂ > 0.76).

Solve a memory test problem involving a sample of 625 subjects, where 80% are expected to pass. Learn how to calculate the mean and standard deviation of the sample proportion, and find the probability of having more than 76% pass the test using z-scores and normal approximation. Suitable for high school statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation