When consumers apply for credit, their credit is rated using the FICO score. A bank manager used a random sample of the credit scores of 18 customers and determined the following 95% confidence interval:
(613.6, 708.0)
If the sample used 45 customers rather than 18 customers, 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 larger sample size.
B) The confidence interval would be very similar since we are still estimating FICO scores which have wide variation.
C) Given the confidence level of 95%, the interval calculated using a sample of 44 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 larger sample size.
A) The confidence interval would be narrower since the standard error would be smaller due to the larger sample size.
The confidence interval is directly affected by the sample size. As the sample size increases, the standard error decreases. This narrower standard error results in a narrower confidence interval, indicating a more precise estimate of the population parameter. Therefore, with a larger sample size of 45 customers, we would expect the 95% confidence interval to be narrower compared to the interval calculated using a sample of 18 customers.
The correct answer is A) The confidence interval would be narrower since the standard error would be smaller due to the larger sample size.
In statistical analysis, the confidence interval is a range of values within which the true population parameter is estimated to lie. A larger sample size generally leads to a more precise estimate of the population parameter and, consequently, a narrower confidence interval.
The standard error is a measure of the variability of the sample mean. As the sample size increases, the standard error decreases, indicating a smaller spread of the sample mean around the population mean. This reduction in standard error results in a narrower confidence interval.
In this case, if the sample size were increased from 18 to 45 customers, you would expect the 95% confidence interval to be narrower because the standard error would be smaller. This means that the estimate of the credit scores would be more precise, providing a narrower range within which the true population mean is expected to fall.