what happens to the width of our confidence interval for mu if we increase the sample size but keep the confidence level the same ?
a. it increases
b. it stays the same
c. it decreases
d. it depends on the sample mean and standard deviation
C. SEm = SD/√n
To understand what happens to the width of the confidence interval for the population mean (mu) when we increase the sample size while keeping the confidence level the same, we need to understand the fundamental concepts of confidence intervals.
A confidence interval is a range of values that provides an estimate of the parameter, in this case, the population mean. The width of the confidence interval indicates the range of values that we are reasonably confident contains the true population mean.
When we increase the sample size, we are obtaining more data points, which adds more information to our analysis. With a larger sample size, we have more precision in estimating the population mean. As a result, the variability in the estimation decreases, leading to a narrower confidence interval.
Therefore, the correct answer is (c) it decreases. Increasing the sample size reduces the width of the confidence interval because more data points lead to a more precise estimate of the population mean.