Suppose x has a normal distribution with σ=6. A random...

Suppose $x$ has a normal distribution with $\sigma=6$. A random sample of size 16 has sample mean 50 . (a) Check Requirements Is it appropriate to use a normal distribution to compute a confidence interval for the population mean $\mu ?$ Explain. (b) Find a $90 \%$ confidence interval for $\mu$. (c) Interpretation Explain the meaning of the confidence interval you computed.

Suppose $x$ has a normal distribution with $\sigma=6$. A random sample of size 16 has sample mean 50 .
(a) Check Requirements Is it appropriate to use a normal distribution to compute a confidence interval for the population mean $\mu ?$ Explain.
(b) Find a $90 \%$ confidence interval for $\mu$.
(c) Interpretation Explain the meaning of the confidence interval you computed.

Key Concepts

Interpretation of a Confidence Interval

The interpretation of a confidence interval is that, if we were to take many random samples and construct a confidence interval from each sample, approximately the given percentage (e.g., 90%) of those intervals would contain the true population mean. This does not mean there is a 90% probability that the true mean lies within any single computed interval, but rather reflects the reliability of the method over many repetitions.

Critical Value

In constructing a confidence interval for the population mean with known population standard deviation, a critical value from the standard normal distribution is used. This critical value corresponds to the desired level of confidence and determines the number of standard errors to extend on either side of the sample mean, forming the boundaries of the interval.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter at a specified confidence level. The interval is typically calculated by taking the sample mean and adding and subtracting the product of the critical value and the standard error, thereby providing a measure of uncertainty around the estimate.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetry about the mean, and specific parameters such as the mean and standard deviation. When data are normally distributed, it simplifies the process of making inferences about the population because many statistical techniques, including confidence intervals, rely on this assumption.

Standard Error

The standard error measures the variability of the sample mean around the population mean and is calculated by dividing the population standard deviation by the square root of the sample size. This concept is crucial because it quantifies how much the sample mean is expected to fluctuate from sample to sample, influencing the width of the confidence interval.

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