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 $n(1-p) \geq 10$).