Question

ch 16 i collect a random sample of size n from a population and from the data collected compute a 95% confidence interval for the mean of the population. which of the following would produce a new confidence interval with smaller width (smaller margin of error) based on these same data? use the same confidence level, but compute the interval n times. approximately 5% of these intervals will be larger. nothing can guarantee that you will get a smaller interval. one can only say the chance of obtaining a smaller interval is 0.05. use a larger confidence level. use a smaller confidence level.

Explanation:

Step1: Recall confidence - interval formula

The formula for a confidence interval for the population mean (when the population standard - deviation $\sigma$ is known) is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, and the width of the confidence interval is $w = 2z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. Here, $z_{\alpha/2}$ is the z - score corresponding to the level of confidence, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Step2: Analyze the relationship between confidence level and $z_{\alpha/2}$

The confidence level $C=1 - \alpha$. As the confidence level decreases, $\alpha$ increases, and $z_{\alpha/2}$ decreases. Since the width of the confidence interval $w$ is directly proportional to $z_{\alpha/2}$ (when $\sigma$ and $n$ are fixed), a smaller $z_{\alpha/2}$ will result in a smaller width of the confidence interval.

Step3: Evaluate each option

• Option 1: Computing the interval $n$ times at the same confidence level does not change the width of the individual intervals. It just gives multiple intervals, and about $(1 - C)$ of them will not contain the true population mean, but it has no effect on the width of a single interval.
• Option 2: There are ways to change the confidence level to guarantee a smaller interval.
• Option 3: Using a larger confidence level will increase $z_{\alpha/2}$, which will increase the width of the confidence interval.
• Option 4: Using a smaller confidence level will decrease $z_{\alpha/2}$, which will decrease the width of the confidence interval.

Answer:

Use a smaller confidence level.