To solve this problem, let's first define what each term represents:

• Sample proportion refers to the proportion of Soft Lilac nail polish bottles in the manager's sample.
• Population proportion refers to the proportion of Soft Lilac nail polish bottles in the entire factory, based on the provided data.

(a) Sample Proportion

The sample proportion $\hat{p}$ is calculated as the ratio of Soft Lilac bottles in the sample to the total number of bottles in the sample.

$\hat{p} = \frac{\text{Number of Soft Lilac bottles in sample}}{\text{Total number of bottles in sample}} = \frac{19}{74}$

Calculating this:

$\hat{p} = \frac{19}{74} \approx 0.26$

Thus, the sample proportion is 0.26.

(b) Population Proportion

The population proportion $p$ is calculated as the ratio of Soft Lilac bottles in the entire population (factory) to the total number of bottles in the population.

$p = \frac{\text{Number of Soft Lilac bottles in population}}{\text{Total number of bottles in population}} = \frac{8}{33}$

Calculating this:

$p = \frac{8}{33} \approx 0.24$

Thus, the population proportion is 0.24.

Final Answer:

(a) Sample proportion: 0.26
(b) Population proportion: 0.24

Do you want further details or have any questions?

Here are 5 related questions to expand on this:

1. How is a sample proportion different from a population proportion?
2. How can we interpret the difference between the sample and population proportions?
3. What is the importance of random sampling in estimating population proportions?
4. What would happen to the sample proportion if we took a larger sample size?
5. How can we calculate confidence intervals for sample proportions?

Tip: Always ensure that sample sizes are large enough to represent the population effectively. Larger samples tend to provide more accurate estimates.