Below, n is the sample size, p is the population proportion and p̂ is the sample proportion. Use the Central Limit Theorem and the TI-84 Plus calculator to find the probability. Round the answers to at least four decimal places. n=200 p=0.10 P(p̂<0.07)=__

To solve \( P(\hat{p} < 0.07) \), we can use the Central Limit Theorem (CLT) for proportions. Here are the steps:

1. **Identify parameters:**
   - Sample size, \( n = 200 \)
   - Population proportion, \( p = 0.10 \)
   
2. **Compute the mean and standard deviation of the sampling distribution for \( \hat{p} \):**
   - Mean \( \mu_{\hat{p}} = p = 0.10 \)
   - Standard deviation \( \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.10 \times 0.90}{200}} \)

   Calculating \( \sigma_{\hat{p}} \):
   \[
   \sigma_{\hat{p}} = \sqrt{\frac{0.10 \times 0.90}{200}} = \sqrt{0.00045} \approx 0.0212
   \]

3. **Convert \( P(\hat{p} < 0.07) \) to a Z-score:**
   - Find the Z-score for \( \hat{p} = 0.07 \):
     \[
     Z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.07 - 0.10}{0.0212} \approx -1.4151
     \]

4. **Find the probability using the Z-score:**
   - Using the TI-84, input the Z-score into the normal cumulative distribution function (`normalcdf`) to find \( P(Z < -1.4151) \).
   - On the TI-84: `normalcdf(-1E99, -1.4151)`.

5. **Conclusion:**
   The probability \( P(\hat{p} < 0.07) \approx 0.0784 \).

So, the final answer is:
\[
P(\hat{p} < 0.07) \approx 0.0784
\]

Would you like additional details on any steps? 

### Related Questions
1. How would you calculate \( P(\hat{p} > 0.07) \) for the same values of \( n \) and \( p \)?
2. What if the sample size, \( n \), increased to 500? How would this affect the probability?
3. How does the Central Limit Theorem apply to sample means compared to sample proportions?
4. What is the probability \( P(\hat{p} = 0.10) \) given this distribution?
5. Can the TI-84 calculator also perform hypothesis testing for proportions?

#### Tip
Always verify that both \( np \) and \( n(1 - p) \) are greater than 5 to ensure the sample proportion approximates a normal distribution under the CLT.

Below, n is the sample size, p is the population proportion and p̂ is the sample proportion. Use the Central Limit Theorem and the TI-84 Plus calculator to find the probability. Round the answers to at least four decimal places. n=200 p=0.10 P(p̂<0.07)=__

Use the Central Limit Theorem and TI-84 calculator to find the probability that a sample proportion p̂ is less than 0.07, given a sample size of n=200 and population proportion p=0.10. Step-by-step calculations include mean, standard deviation, and Z-score, with answers rounded to four decimal places. Ideal for high school and college 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