A random sample has been taken from a normal distribution...

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02,61.98) and (39.95,60.05) (a) What is the value of the sample mean? (b) One of these intervals is a $95 \% \mathrm{CI}$ and the other is a $90 \%$ CI. Which one is the $95 \%$ CI and why?

A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02,61.98) and (39.95,60.05)
(a) What is the value of the sample mean?
(b) One of these intervals is a $95 \% \mathrm{CI}$ and the other is a $90 \%$ CI. Which one is the $95 \%$ CI and why?

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Key Concepts

Confidence Interval

A confidence interval provides a range of values that is likely to contain the population parameter of interest, based on sample data and a specified level of confidence. It is calculated using the sample statistic and is adjusted using a critical value from the appropriate statistical distribution.

Confidence Level

The confidence level represents the probability that the calculated interval from repeated samples will contain the true population parameter. Higher confidence levels require using larger critical values, which in turn widen the interval, reflecting more conservative estimation.

Sample Mean

The sample mean is the arithmetic average of the observations in the sample and serves as an unbiased estimator of the population mean. In symmetric distributions like the normal distribution, the sample mean is located at the center of the confidence interval.

Interval Width and Critical Value

The width of a confidence interval is influenced by the critical value from the statistical table (such as the z or t distribution) and the standard error of the estimate. A larger critical value, which corresponds to a higher confidence level, results in a wider interval, while a smaller critical value gives a narrower interval.

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