Smartphones: A poll agency reports that 48% of teenagers aged 12-17 own smartphones. A random sample of 150 teenagers is drawn. Round your answers to four decimal places as needed. (e) Find the probability that less than 55% of sampled teenagers own smartphones.

To find the probability that less than 55% of the sampled teenagers own smartphones, we can use the normal approximation to the binomial distribution since the sample size \( n = 150 \) is large. Here’s how we proceed:

1. **Given Values and Setup**:
   - Population proportion \( p = 0.48 \)
   - Sample size \( n = 150 \)
   - We are looking for the probability that the sample proportion \( \hat{p} \) is less than 0.55.

2. **Calculate the Mean and Standard Deviation of the Sampling Distribution**:
   - The mean of the sampling distribution for \( \hat{p} \) is \( \mu_{\hat{p}} = p = 0.48 \).
   - The standard deviation of the sampling distribution for \( \hat{p} \) is:
     \[
     \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.48 \times (1 - 0.48)}{150}}.
     \]

   Calculating \( \sigma_{\hat{p}} \):
   \[
   \sigma_{\hat{p}} = \sqrt{\frac{0.48 \times 0.52}{150}} = \sqrt{\frac{0.2496}{150}} \approx 0.0408.
   \]

3. **Convert to a Z-Score**:
   - We want \( P(\hat{p} < 0.55) \), which translates to finding the probability that the sample proportion is less than 0.55.
   - Convert \( 0.55 \) to a z-score using the formula:
     \[
     z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.55 - 0.48}{0.0408}.
     \]

   Calculating the z-score:
   \[
   z = \frac{0.55 - 0.48}{0.0408} \approx 1.716.
   \]

4. **Find the Probability**:
   - Now, we look up the z-score of \( 1.716 \) in the standard normal distribution table or use a calculator. The cumulative probability for \( z = 1.716 \) is approximately \( 0.9564 \).

Therefore, the probability that less than 55% of the sampled teenagers own smartphones is approximately:

\[
P(\hat{p} < 0.55) \approx 0.9564.
\]

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### Related Questions
1. How would the probability change if the sample size were increased to 200?
2. What is the probability that more than 60% of sampled teenagers own smartphones?
3. How does the normal approximation apply to smaller sample sizes?
4. If the population proportion were 50%, how would that affect the result?
5. How do we calculate the probability for exactly 55% of teenagers owning smartphones?

#### Tip:
For problems involving sample proportions, use the normal approximation to the binomial distribution when \( np \) and \( n(1 - p) \) are both greater than 5.

A poll agency reports that 48% of teenagers aged 12-17 own smartphones. A random sample of 150 teenagers is drawn. Find the probability that less than 55% of sampled teenagers own smartphones.

Calculate the probability that less than 55% of a random sample of teenagers own smartphones using the normal approximation to the binomial distribution. This example covers probability, sampling distribution, and z-score calculation for a sample size of 150, with a population proportion of 48%.

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