Answer:
a. To find the planning value for the population standard deviation (σ), you generally have two options:
1. If you have access to historical data or some reasonable estimate for the population standard deviation, you can use that value. This would be the best approach if you have relevant information specific to your population.
2. If you don't have access to any specific information about the population standard deviation, you can use a common rule of thumb, which is to estimate σ as 1/4th of the range (the difference between the maximum and minimum values) of the potential salaries. In this case, the range is $45,000 - $10,000 = $35,000. So, σ = 1/4 * $35,000 = $8,750.
b. To determine the sample size (n) needed for a desired margin of error (E), you can use the formula:
\[n = \frac{{(Z * σ)^2}}{{E^2}}\]
Where:
- Z is the critical value corresponding to the desired confidence level (for a 95% confidence interval, Z is approximately 1.96).
- σ is the estimated population standard deviation.
- E is the desired margin of error.
Let's calculate the sample size for a margin of error of $500 using both the estimated σ from option a and the larger σ based on the range of salaries:
For σ = $8,750:
\[n = \frac{{(1.96 * 8,750)^2}}{{500^2}} \approx 138.22\]
Since you cannot have a fraction of a person in your sample, you would need a sample size of at least 139.
For σ = $17,500 (1/4th of the range):
\[n = \frac{{(1.96 * 17,500)^2}}{{500^2}} \approx 554.88\]
Again, you cannot have a fraction of a person in your sample, so you would need a sample size of at least 555.
c. The choice of whether to obtain a sample size of 80 or 150 depends on your specific goals and resources. Here are some considerations:
- **Cost and Resources:** A larger sample size (150) would require more time and resources for data collection and analysis compared to a smaller sample size (80).
- **Precision:** A larger sample size generally results in a smaller margin of error, which means a more precise estimate of the population mean. If obtaining a highly accurate estimate of the population mean is crucial, you may opt for the larger sample.
- **Practicality:** If obtaining a sample size of 150 is challenging due to budget or time constraints, you might have to settle for a sample size of 80. In this case, you should be aware that the margin of error will be larger, and your confidence in the estimate may be slightly lower.
In summary, the choice between 80 and 150 as the sample size depends on your specific needs, constraints, and the level of precision required for your study.
Explanation: