Hoping to lure more shoppers downtown, a city builds a new parking garage in the central business district. The city hopes that the parking revenues will pay for the garage.
During a recent two month period (44 days), the city statistician calculated the 95% confidence interval for the mean daily parking fees collected as:
($121, $131)
If the sample used 22 days rather than 44 days, how would you expect the 95% confidence interval to change? Support your answer.
A) The confidence interval would be narrower since the standard error would be smaller due to the smaller sample size.
B) The confidence interval would be very similar since we are still estimating parking rate income.
C) Given the confidence level of 95%, the interval calculated using a sample of 22 would be about the same since the confidence is not very high.
D) The confidence interval would be wider since the standard error would be larger due to the smaller sample size.
Standard error = .385/sqrt ((44)) = 0.058
E = 1.96 *. 385 sqrt(44) = 5.00
Mean - E, Mean + E
126-5, 126 + 5
($121, $131)
Standard error = .385/sqrt ((22) = 0.082
E = 1.96 * .385 sqrt(22) = 3.5
Mean - E, Mean + E
126- 3.5, 126 + 3.5
( $122.5, $129.5)
To answer this question, we need to understand how the confidence interval is calculated and how sample size affects it.
A confidence interval is a range of values within which we can be confident that the true population parameter lies. In this case, the population parameter is the mean daily parking fees collected.
The formula to calculate the confidence interval for the mean is:
CI = sample mean ± (critical value) * (standard error)
The critical value is based on the desired confidence level and the sample size. In this case, the confidence level is 95%.
The standard error is a measure of the variability of the sample mean. It depends on the standard deviation of the population and the sample size.
Now, let's compare the effect of changing the sample size from 44 days to 22 days on the confidence interval.
When the sample size decreases, the standard error increases. This is because with a smaller sample, there is more uncertainty in estimating the true population parameter.
Since the standard error is used to calculate the width of the confidence interval, increasing the standard error will result in a wider interval.
Therefore, the correct answer is:
D) The confidence interval would be wider since the standard error would be larger due to the smaller sample size.
By having a smaller sample size, the city would have more uncertainty in estimating the mean daily parking fees, resulting in a wider confidence interval.