A random sample of size 36 is drawn from an x distribution....

A random sample of size 36 is drawn from an $x$ distribution. The sample mean is 100 . (a) Suppose the $x$ distribution has $\sigma=30$. Compute a $90 \%$ confidence interval for $\mu$. What is the value of the margin of error? (b) Suppose the $x$ distribution has $\sigma=20$. Compute a $90 \%$ confidence interval for $\mu$. What is the value of the margin of error? (c) Suppose the $x$ distribution has $\sigma=10$. Compute a $90 \%$ confidence interval for $\mu$. What is the value of the margin of error? (d) Compare the margins of error for parts (a) through (c). As the standard deviation decreases, does the margin of error decrease? (e) Critical Thinking Compare the lengths of the confidence intervals for parts (a) through (c). As the standard deviation decreases, does the length of a $90 \%$ confidence interval decrease?

A random sample of size 36 is drawn from an $x$ distribution. The sample mean is 100 .
(a) Suppose the $x$ distribution has $\sigma=30$. Compute a $90 \%$ confidence interval for $\mu$. What is the value of the margin of error?
(b) Suppose the $x$ distribution has $\sigma=20$. Compute a $90 \%$ confidence interval for $\mu$. What is the value of the margin of error?
(c) Suppose the $x$ distribution has $\sigma=10$. Compute a $90 \%$ confidence interval for $\mu$. What is the value of the margin of error?
(d) Compare the margins of error for parts (a) through (c). As the standard deviation decreases, does the margin of error decrease?
(e) Critical Thinking Compare the lengths of the confidence intervals for parts (a) through (c). As the standard deviation decreases, does the length of a $90 \%$ confidence interval decrease?

Key Concepts

Standard Error

The standard error quantifies the variability of the sample mean estimate from the true population mean. It is calculated by dividing the standard deviation by the square root of the sample size, and it indicates how precise the sample mean is as an estimate of the true value.

Standard Deviation

Standard deviation measures the spread or dispersion of observations in a distribution around the mean. In the context of confidence intervals, a smaller standard deviation leads to a smaller standard error and thus a narrower confidence interval, indicating a more precise estimate of the population parameter.

Critical Value (z-value)

The critical value is a point on the standard normal distribution that corresponds to the desired confidence level of the interval. For a 90% confidence level, this value is determined so that 90% of the distribution lies between the negative and positive of this value, and it plays a key role in determining the width of the interval.

Margin of Error

The margin of error is the term added and subtracted from the sample mean to create the confidence interval. It represents the maximum expected difference between the true population parameter and a sample estimate, and is computed as the product of the critical value and the standard error, decreasing with lower variability or larger sample sizes.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. It is constructed using the sample mean and a margin of error, and the interval reflects the uncertainty inherent in estimating the parameter from a finite sample.

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