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.