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?
How planning value is calculated and where the 4 come from?
planning value = (45000-30000)/4 = 15000/4 = 3750
a. $500?
n = (Za/2* σ/E)^2
n = (1.96* 3750/500)^2
n = 217
b. $200?
n = (Za/2* σ/E)^2
n = (1.96*3750/200)^2
n = 1351
c. $100?
n = (Za/2* σ/E)^2
n = (1.96* 3750/100)^2
n = 5403
What is the planning value for the population standard deviation (to the nearest whole number)?
3750
How large a sample should be taken if the desired margin of error is as shown below (to the nearest whole number)?
a. $500?
216
b. $200?
1351
c. $100?
5403
d. Would you recommend trying to obtain the $100 margin of error?
No, sample requires too much time and expense
A Phoenix Wealth Management/Harris Interactive survey of 1500 individuals with net worth of $1 million or more provided a variety of statistics on wealthy people (BusinessWeek, September 22, 2003). The previous three-year period had been bad for the stock market, which motivated some of the questions asked.
a. The survey reported that 53% of the respondents lost 25% or more of their portfolio value over the past three years. Develop a 95% confidence interval for the proportion of wealthy people who lost 25% or more of their portfolio value over the past three years. (to 3 decimals)
You use 4 as a 'best guess' when you don't have a previous study's standard deviation or haven't used a pilot study's standard deviation. It is a widely accepted rule in Statistics when the first two options are not available.
the 4 comes from the standard deviation being exactly 1/4 of the range.
To determine the planning value for the population standard deviation, we need additional information. It is not explicitly provided in your question.
Moving on to the sample size calculation, the formula to determine the sample size required for estimating a population mean is:
n = (Z * σ / E)^2
Where:
- n is the sample size
- Z is the z-score corresponding to the desired level of confidence (95% confidence level corresponds to a z-score of approximately 1.96)
- σ is the population standard deviation (which we currently don't have)
- E is the desired margin of error (in this case, $500, $200, or $100)
Since the population standard deviation (σ) is not given, we need to estimate it using the formula:
σ ≈ (maximum value - minimum value) / (4 * z)
For this case, the maximum value is $45,000, the minimum value is $30,000, and z is 1.96.
Calculating σ:
σ ≈ (45000 - 30000) / (4 * 1.96)
σ ≈ 15000 / 7.84
σ ≈ 1913.26 (approximately)
Now that we have the estimated population standard deviation, we can proceed to calculate the sample size using the specified margin of error.
a. Margin of error = $500:
n = (1.96 * 1913.26 / 500)^2
b. Margin of error = $200:
n = (1.96 * 1913.26 / 200)^2
c. Margin of error = $100:
n = (1.96 * 1913.26 / 100)^2
By plugging these values into the formulas, you can calculate the sample sizes for each scenario, rounding to the nearest whole number.