Suppose x has a mound-shaped distribution with σ=9. A random...

Suppose $x$ has a mound-shaped distribution with $\sigma=9$. A random sample of size 36 has sample mean 20 . (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 $95 \%$ confidence interval for $\mu$. (c) Interpretation Explain the meaning of the confidence interval you computed.

Suppose $x$ has a mound-shaped distribution with $\sigma=9$. A random sample of size 36 has sample mean 20 .
(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 $95 \%$ confidence interval for $\mu$.
(c) Interpretation Explain the meaning of the confidence interval you computed.

Key Concepts

Central Limit Theorem

This theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the underlying population distribution. In cases where the population is already 'mound-shaped' or approximately normal, as given in the problem, the sample mean is normally distributed even for moderate sample sizes.

Confidence Interval for the Mean

A confidence interval for the population mean is an interval estimate calculated from the sample statistics. When the population standard deviation is known and the sample size is sufficiently large or the population is normally distributed, the z-distribution is used to calculate the interval, typically in the form of sample mean plus or minus the critical value times the standard error.

Standard Error

The standard error is the measure of the variability of the sample mean, calculated as the population standard deviation divided by the square root of the sample size. It quantifies how much the value of the sample mean is expected to vary from sample to sample due to random sampling error.

Normal Distribution Assumptions

Using the normal distribution for confidence interval construction requires that the underlying population distribution is either normally distributed or that the sample size is large enough for the Central Limit Theorem to apply. A 'mound-shaped' distribution implies symmetry and a single peak, often indicating that normal-based methods are appropriate.

Interpretation of Confidence Interval

The interpretation of a confidence interval, such as a 95% confidence interval, is that if the sampling process were repeated many times, approximately 95% of the computed intervals would contain the true population mean. It reflects the degree of confidence one has in the interval as an estimate of the actual mean.

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