The travel-to-work time for residents of the 15 largest cities in the United States is reported in the 2003 Information Please Almanac. Suppose that a preliminary simple random sample of residents of San Francisco is used to develop a planning value of 6.35 minutes for the population standard deviation.
a. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes, what sample size should be used? Assume 95% confidence.
b. If we want to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute, what sample size should be used? Assume 95% confidence.
Formula:
n = {[(z-value)(sd)]/E}^2
a) n = [(1.96 * 6.35)/2]^2
b) n = [(1.96 * 6.35)/1)^2
Calculate. Round to the next highest whole number.
To calculate the sample size required using the margin of error, you can use the formula:
n = (Z * σ / E)^2
Where:
n = required sample size
Z = Z-score (corresponding to the desired confidence level)
σ = population standard deviation
E = margin of error
Since we want a 95% confidence level, the Z-score corresponding to this level is 1.96 (from the standard normal distribution table).
a. For a margin of error of 2 minutes:
n = (1.96 * 6.35 / 2)^2
n = (12.446 / 2)^2
n = 6.223^2
n ≈ 38.76
Therefore, a sample size of 39 residents should be used to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes.
b. For a margin of error of 1 minute:
n = (1.96 * 6.35 / 1) ^ 2
n = (12.446)^2
n ≈ 154.85
Therefore, a sample size of 155 residents should be used to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute.
To estimate the sample size needed for estimating the population mean travel-to-work time for San Francisco residents with a given margin of error, we can use the formula:
n = (Z * σ / E)^2
Where:
- n is the required sample size
- Z is the Z-score corresponding to the desired confidence level
- σ is the population standard deviation
- E is the desired margin of error
a. For a margin of error of 2 minutes and a 95% confidence level, the Z-score is approximately 1.96 (corresponding to a 95% confidence level).
Plugging the values into the formula:
n = (1.96 * 6.35 / 2)^2
n ≈ (12.486 / 2)^2
n ≈ 6.243^2
n ≈ 38.997 ≈ 39
Therefore, a sample size of at least 39 is needed to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 2 minutes and a 95% confidence level.
b. For a margin of error of 1 minute and a 95% confidence level, the Z-score remains the same (approximately 1.96).
Plugging the values into the formula:
n = (1.96 * 6.35 / 1)^2
n ≈ (12.486 / 1)^2
n ≈ 12.486^2
n ≈ 155.74 ≈ 156
Therefore, a sample size of at least 156 is needed to estimate the population mean travel-to-work time for San Francisco residents with a margin of error of 1 minute and a 95% confidence level.