Assume a normal distribution with known population variance....

Assume a normal distribution with known population variance. Calculate the width to estimate the population mean, $\mu$, for the following.
a. $90 \%$ confidence level; $n=100 ; \sigma^2=169$
b. $95 \%$ confidence level; $n=120 ; \sigma=25$

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, which means that it has a bell-shaped curve. It is characterized by its mean and variance (or standard deviation) and is fundamental in statistics because of the central limit theorem, which allows its use in approximating the distribution of sample means.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter with a given level of confidence (such as 90% or 95%). It is typically expressed as the point estimate ± a margin of error, providing a measure of uncertainty around the estimate.

Known Population Variance

When the population variance is known, the standard normal (Z) distribution is used to calculate the confidence interval for the mean. This contrasts with situations where the variance is unknown, in which case a t-distribution is typically used. Knowing the variance simplifies the calculation and reduces uncertainty in the interval estimation.

Z-score

A Z-score quantifies the number of standard deviations a data point is from the mean in a standard normal distribution. In the context of confidence intervals, Z-scores corresponding to the desired confidence level (such as 1.645 for 90% or 1.96 for 95%) determine the width of the interval.

Margin of Error

The margin of error is the product of the appropriate Z-score and the standard error of the sample mean. It defines the radius of the confidence interval, indicating how much the estimate might deviate from the true population mean, thus directly affecting the precision of the estimate.

Sample Size

Sample size plays a crucial role in the precision of the estimate of a population parameter. A larger sample size reduces the standard error, which in turn narrows the confidence interval, leading to a more precise estimate of the population mean.

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