Annual starting salaries for college graduates with degrees in business administration are generally expected to be between $30,000 and $45,000. Assume that a 95% confidence interval estimate of the population mean annual starting salary is desired. What is the planning value for the population standard deviation (to the nearest whole number)?
How large a sample should be taken if the desired margin of error is as shown below (to the nearest whole number)?
a. $500?
b. $200?
c. $100?
To determine the planning value for the population standard deviation, we can use the range provided, which is $30,000 to $45,000.
1. Calculate the range:
Range = Maximum Value - Minimum Value
Range = $45,000 - $30,000
Range = $15,000
2. Divide the range by 4 to estimate the population standard deviation:
Population Standard Deviation ≈ Range / 4
Population Standard Deviation ≈ $15,000 / 4
Population Standard Deviation ≈ $3,750
Therefore, the planning value for the population standard deviation, to the nearest whole number, is $3,750.
To determine the sample size needed with the desired margin of error, we can use the formula:
Sample Size = (Z * σ / E)^2
Where:
Z is the Z-score corresponding to the desired confidence level (in this case, for a 95% confidence level, the Z-score is approximately 1.96).
σ is the estimated population standard deviation.
E is the desired margin of error.
a. For a desired margin of error of $500:
Sample Size = (1.96 * 3750 / 500)^2
Sample Size ≈ (1.96 * 7.5)^2
Sample Size ≈ 14.7^2
Sample Size ≈ 216
Therefore, a sample size of approximately 216 should be taken if the desired margin of error is $500.
b. For a desired margin of error of $200:
Sample Size = (1.96 * 3750 / 200)^2
Sample Size ≈ (1.96 * 18.75)^2
Sample Size ≈ 36.6^2
Sample Size ≈ 1340
Therefore, a sample size of approximately 1340 should be taken if the desired margin of error is $200.
c. For a desired margin of error of $100:
Sample Size = (1.96 * 3750 / 100)^2
Sample Size ≈ (1.96 * 37.5)^2
Sample Size ≈ 73.5^2
Sample Size ≈ 5392
Therefore, a sample size of approximately 5392 should be taken if the desired margin of error is $100.