Let's analyze each part of this exercise one by one.

Part (a)

The question asks if it is appropriate to use the normal approximation to find the probability that less than 6.5% of the sample holds multiple jobs, given $n = 75$ and $p = 0.05$. The rule of thumb for using a normal approximation is if both $np \geq 10$ and $n(1 - p) \geq 10$.

For this problem:

1. $np = 75 \times 0.05 = 3.75$, which is less than 10.
2. $n(1 - p) = 75 \times 0.95 = 71.25$, which is greater than 10.

Since $np < 10$, it is not appropriate to use the normal approximation.

Part (b)

For this part, we use a sample size of $n = 350$ and need to find the probability that less than 6.5% of this sample holds multiple jobs.

Now, since $np = 350 \times 0.05 = 17.5$ and $n(1 - p) = 350 \times 0.95 = 332.5$, both conditions are met to use the normal approximation. We can now find the probability that $\hat{p} < 0.065$ using the normal approximation with:

• Mean $\mu = p = 0.05$
• Standard deviation $\sigma = \sqrt{\frac{p(1 - p)}{n}}$

Substitute these values and calculate the Z-score for 0.065, then find the corresponding probability.

Part (c)

This part asks for the probability that more than 6% of the sample of 350 holds multiple jobs. Using the same normal approximation as in part (b), calculate the probability $\hat{p} > 0.06$.

Part (d)

We are tasked with finding the probability that the proportion of individuals in the sample of 350 who hold multiple jobs is between 0.05 and 0.10. Using the normal distribution calculated in parts (b) and (c), find the probabilities for both 0.05 and 0.10, then take the difference between them.

Part (e)

This part asks if it would be unusual for less than 4% of the sample of 350 to hold multiple jobs. This can be evaluated by calculating the probability $\hat{p} < 0.04$. If this probability is very low (typically less than 0.05), it would be considered unusual.

Let me know if you'd like detailed calculations or further explanations on any of these parts.

Additional Questions:

1. How would the solution differ if the sample size was changed to 150 instead of 75 or 350?
2. What impact does increasing the population proportion (e.g., from 5% to 10%) have on the appropriateness of the normal approximation?
3. How would you calculate the probability of a range if the normal approximation was not valid?
4. Why is the normal approximation preferred in cases with large sample sizes?
5. How does changing the percentage threshold (like from 6% to 8%) affect the probability in part (c)?

Tip:

When using the normal approximation, always verify the conditions $np \geq 10$ and $n(1 - p) \geq 10$ to ensure accurate results.