Confidence Interval for a Mean (with known standard deviation)

In previous chapters, we've known the mean value of a normal distribution and, based on the z-table values, said that 90% of the observations will fall +/- 1.65 standard deviations from the mean. Similarly, 95% of the observations will fall +/- 1.96 standard deviations from the mean.

In this chapter, we invert the logic. We say that, given a sample mean, there is a 90% confidence level that the population mean is within +/- 1.65 standard errors of the mean from the sample mean. And similarly, there is a 95% confidence level that the population mean lies withing +/- 1.96 standard errors of the mean from the sample mean.

This type of construction of a confidence interval around a sample mean requires us to know the standard deviation of the population (sigma), from which we calculate the standard error of the mean, sigma/sqrt(n), where n is the sample size. It's not very realistic that we would know the standard deviation of the population and not know its mean. So this is just an exercise. Later in chapter 8, we'll deal with cases where we don't know either the mean or the std dev of the population.

Some terminology

If we want a x% confidence level, we call that 1-alpha.

So if we want a 99% confidence level, 1-alpha = 99% = 0.99. And alpha = 0.01.

Typically, we're looking for a symmetric interval around the sample mean. So in this case we want to know that the population mean falls within 45.5% on either side of the sample mean. Therefore, we look at alpha/2 on each side. We look up z_{alpha/2} in the table to find the z-score such that the area in the tail of the curve is alpha/2. Note: this actually means you find the value for 0.5-alpha/2 in the table.

Now we can construct our confidence interval as:

Some common values for Z_{alpha/2}:

90% confidence interval: Z_{0.05} = 1.65

95% confidence interval: Z_{0.025} = 1.96

99% confidence interval: Z_{0.005} = 2.58

A larger level of confidence requires a larger interval.

I've noted in a prior post that the term "margin of error" seems to be used for the +/- term in the confidence interval in the lecture, but that term is not used at all in the textbook.

## Sunday, February 10, 2008

### Lecture 5 - Ch 8 - Confidence Interval Estimation

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