To calculate the sample size needed to estimate the mean number of miles within a certain margin of error, the z-score associated with the desired confidence level, the population standard deviation, and the desired margin of error need to be known. Here are the steps to calculate the sample size:
1. The confidence level is 95%. For a 95% confidence interval, the z-score (found in Z-table or can be calculated) is approximately 1.96.
2. The population standard deviation σ (sigma), is given as 1000 miles.
3. The desired margin of error (E) is given as 400 miles.
4. Use the formula for sample size: n = (Z * σ / E) ^2 , where 'n' is the sample size, 'Z' is the z-score for our confidence level, 'σ' is the population standard deviation, and 'E' is the margin of error.
Substituting the values into the formula generates:
n = (1.96 * 1000 / 400) ^ 2 = 24.01
Since we cannot have a fraction of a car, we will need to round up to the nearest whole number. The sample size required is 25.
Therefore, the researcher would need a size of 25 cars in the sample to be 95% confident that the mean number of miles is estimated within a margin of error of 400 miles.