Question

1. just some general questions:

a) increasing my confidence level will __________ the margin of error and therefore will also __________ the width of my confidence interval.
b) increasing my sample size will __________ the margin of error and therefore will also __________ the width of my confidence interval.
c) do you think we can perform a 100% confidence interval? how about 99.999% c.i.?

Explanation:

Step1: Margin of error formula

The margin of error \(E = z\times\frac{\sigma}{\sqrt{n}}\) (for population mean, when \(\sigma\) is known). The confidence interval is \(\bar{x}\pm E\), so its width is \(2E\).

Step2: Effect of confidence level

As the confidence level increases, the \(z -\) value (critical value) increases. Since \(E\propto z\) (when \(\sigma\) and \(n\) are constant), increasing the confidence level increases \(E\). And since the width of the confidence interval is \(2E\), increasing \(E\) increases the width of the confidence interval.

Step3: Effect of sample size

As \(n\) increases, since \(E\propto\frac{1}{\sqrt{n}}\) (when \(\sigma\) and \(z\) are constant), increasing \(n\) decreases \(E\). And since the width of the confidence interval is \(2E\), decreasing \(E\) decreases the width of the confidence interval.

Step4: 100% confidence interval

A \(100\%\) confidence interval would require \(z
ightarrow\infty\), which is not practical. For a \(99.999\%\) confidence interval, we can find a very large \(z -\) value (from the standard normal distribution table or using statistical software) and calculate the interval, but it will be very wide.

Answer:

a) Increasing my confidence level will increase the margin of error and therefore will also increase the width of my confidence interval.
b) Increasing my sample size will decrease the margin of error and therefore will also decrease the width of my confidence interval.
c) We cannot perform a \(100\%\) confidence interval in a practical sense (the interval would be infinitely wide). We can perform a \(99.999\%\) confidence interval (by finding the appropriate critical value \(z\) and using the formula for the confidence interval), but it will be a very wide interval.