Construct the appropriate confidence interval. A simple...

Construct the appropriate confidence interval.
A simple random sample of size $n=22$ is drawn from a population that is normally distributed with $\sigma=37 .$ The sample mean is found to be $\bar{x}=122.5 .$ Construct a $90 \%$ confidence interval about the population mean.

Key Concepts

Confidence Interval

A confidence interval is a range of values used to estimate an unknown population parameter with a given level of confidence. It provides an interval within which the true parameter is expected to lie, based on the sample data and the chosen confidence level.

Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic, such as the sample mean, derived from repeated samples of the same size from the population. It is fundamental for inference because it allows us to understand how the statistic varies from sample to sample.

Standard Error

The standard error is a measure of the variability or dispersion of the sampling distribution of a statistic, typically calculated as the population standard deviation divided by the square root of the sample size. It quantifies the uncertainty associated with the sample estimate of the population parameter.

Z-Critical Value

The z-critical value is a cutoff point on the standard normal distribution that corresponds to the desired level of confidence. It is used to determine the margin of error in confidence interval calculations when the population standard deviation is known.

Margin of Error

The margin of error is the product of the z-critical value and the standard error, representing the maximum expected difference between the observed sample statistic and the true population parameter at the given level of confidence. It determines the half-width of the confidence interval.

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