A dean wishes to estimate the average cost of the freshman year at a particular college correct to within $500, with a probability of 0.95. If a random sample of freshmen is to be selected and each asked to keep financial data, how many must be included in the sample? Assume that the dean knows only that the range of expenditures will vary from approximately $4800 to $13000.
0.95 confidence find the confidence interval for the mean of the whole population
To estimate the average cost of the freshman year at a particular college with a desired level of precision, we can use the formula for sample size calculation:
n = (Z * σ / E)²
Where:
n = sample size
Z = Z-score (corresponding to the desired level of confidence)
σ = standard deviation of the population (unknown in this case)
E = desired margin of error
In this case, the dean wants to estimate the average cost to within $500 with a probability of 0.95. The margin of error (E) represents half the range ($500 / 2 = $250) since the range of expenditures is given as $4800 to $13000.
The Z-score corresponding to a 95% confidence level is approximately 1.96. We can use this value assuming the sample size is large enough.
Now, substituting the given values into the formula:
n = (1.96 * σ / 250)²
The standard deviation (σ) is unknown, but since we don't have any information other than the range of expenditures ($4800 to $13000), we can use the range rule of thumb to estimate it.
The range rule of thumb states that for a relatively large sample size, the range can be approximated as 6 times the standard deviation (range ≈ 6 * σ). In this case, the range is approximately $8200 ($13000 - $4800).
So, we can estimate the standard deviation as σ ≈ range / 6 ≈ $8200 / 6 ≈ $1367.
Substituting the estimated standard deviation into the formula:
n = (1.96 * 1367 / 250)²
Calculating this expression will give us the required sample size.