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For **Part 2 of 6**, you are asked to find the **standard deviation** (\(\sigma_{\hat{p}}\)) of the sampling distribution for the proportion of teenagers who own smartphones.

The formula for the standard deviation of the sampling distribution of a sample proportion is:

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

Where:
- \(p\) is the population proportion (in this case, \(p = 0.48\), or 48%),
- \(n\) is the sample size (given as \(n = 150\)).

Now, let's calculate it:

\[
\sigma_{\hat{p}} = \sqrt{\frac{0.48(1 - 0.48)}{150}} = \sqrt{\frac{0.48 \times 0.52}{150}} = \sqrt{\frac{0.2496}{150}} = \sqrt{0.001664} \approx 0.0408
\]

Thus, the standard deviation \(\sigma_{\hat{p}}\) is approximately **0.0408**.

Would you like further explanation or details?

Here are 5 related questions you might consider:
1. What is the significance of the standard deviation in a sampling distribution?
2. How does increasing the sample size affect the standard deviation?
3. How do you interpret the value of \(\sigma_{\hat{p}}\)?
4. Can the standard deviation of a sample proportion be larger than the population proportion itself?
5. Why is it important to use a random sample when calculating statistics like this?

**Tip:** Always remember to check whether your sample size \(n\) is large enough to meet the conditions for normal approximation when using proportions (i.e., \(np \geq 10\) and \(n(1-p) \geq 10\)).

Find the standard deviation (σ̂p) for a sample proportion, given a population proportion (p = 0.48) and sample size (n = 150).

Learn how to calculate the standard deviation for a sample proportion with a population proportion of 0.48 and a sample size of 150. Step-by-step solution included, using the formula σ̂p = √(p(1 - p) / n). Suitable for Grades 11-12 statistics students.

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