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To solve this problem, we need to find the probability that the sample proportion \(\hat{p}\) falls between 0.11 and 0.19.

We will use the normal approximation to the binomial distribution to find this probability, given that the sample size \(n\) is large.

Here are the steps:

1. **Calculate the mean of the sampling distribution of \(\hat{p}\)**:
   \[
   \mu_{\hat{p}} = p = 0.14
   \]

2. **Calculate the standard deviation of the sampling distribution of \(\hat{p}\)**:
   \[
   \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.14 \times 0.86}{148}}
   \]

3. **Standardize the bounds (0.11 and 0.19) using the z-score formula**:
   \[
   z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}
   \]

4. **Find the z-scores for 0.11 and 0.19**:
   \[
   z_{0.11} = \frac{0.11 - 0.14}{\sigma_{\hat{p}}}
   \]
   \[
   z_{0.19} = \frac{0.19 - 0.14}{\sigma_{\hat{p}}}
   \]

5. **Use the z-scores to find the corresponding probabilities from the standard normal distribution**:
   \[
   P(0.11 < \hat{p} < 0.19) = P(z_{0.11} < Z < z_{0.19})
   \]

Let's compute these values step-by-step.

### Step-by-Step Calculation

1. **Calculate the standard deviation**:
   \[
   \sigma_{\hat{p}} = \sqrt{\frac{0.14 \times 0.86}{148}} = \sqrt{\frac{0.1204}{148}} = \sqrt{0.0008135} \approx 0.0285
   \]

2. **Calculate the z-scores**:
   \[
   z_{0.11} = \frac{0.11 - 0.14}{0.0285} = \frac{-0.03}{0.0285} \approx -1.0526
   \]
   \[
   z_{0.19} = \frac{0.19 - 0.14}{0.0285} = \frac{0.05}{0.0285} \approx 1.7544
   \]

3. **Find the probabilities corresponding to the z-scores**:
   Using the standard normal distribution table or a calculator:
   \[
   P(Z < -1.0526) \approx 0.1462
   \]
   \[
   P(Z < 1.7544) \approx 0.9602
   \]

4. **Calculate the probability**:
   \[
   P(0.11 < \hat{p} < 0.19) = P(Z < 1.7544) - P(Z < -1.0526) = 0.9602 - 0.1462 = 0.8140
   \]

Therefore, the probability \(P(0.11 < \hat{p} < 0.19)\) is approximately **0.8140**.

Would you like any further details or have any questions?

Here are some additional questions you might want to ask:
1. How is the standard deviation of the sample proportion calculated?
2. What are the conditions for using the normal approximation to the binomial distribution?
3. Can you explain the steps to find the z-score?
4. How do you interpret the probability result in the context of this problem?
5. What are some common uses of the normal approximation in statistics?

**Tip:** Always check the conditions for the normal approximation (np and n(1-p) should both be greater than 5) before using it for binomial probabilities.

Learn how to calculate the probability using normal approximation to the binomial distribution for a sample proportion falling between 0.11 and 0.19. Step-by-step calculation of mean, standard deviation, z-scores, and final probability. Suitable for advanced college-level statistics courses.

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