Confidence Intervals for Means


A confidence interval can be thought of as a range that conains a population parameter (such as the population mean) a given percentage of the time.

So, if I say that we have a 95% CI for a population mean that is between 56 and 58, I am saying that 95% of the time, I expect the true population mean to be between 56 and 58.

Remember that it is often impossible and impractical to calculate a true population parameter. Imagine that you needed to know the exact mean (average) weight for all females in the world (the entire population of females). How can you get this mean weight? It is not likely that you will have the time, money, or resources to weigh all woman, collect all data, and calculate the true mean (the true population parameter).

However, you can use sampling to estimate it. The larger the sample and the more samples you collect, the better the estimate.

A confidence interval is an interval (range) that estimates the population parameter – such as a population mean.

Example 1:

Estimate the population mean for adult female height using a 95% confidence interval. The sample mean (x bar) for this example is 65 inches. The sample size (n) is 100, and the sample standard deviation is 2 inches. Note that if you have any sample of data, you can calculate your own sample mean and sample standard deviation, as well as the sample size.

The FORMULA for the Confidence Interval (CI) for “means” is:
sample mean – E   < population mean < sample mean + E

E = (zc * s)/sqrt(n)

To calculate the CI, you must calculate “E” (the error).

To calculate “E”, you need zc:

“zc” which is the cumulative z value (from any z table) for the CI percentage given. In this case, our CI percentage is 95%.

To get zc:

  • 95% is .95
  • 1 – .95 = .05   (so we have .05 in BOTH tails)
  • .05/2 = .025 (in each tail)
  • 1 – .025 = .975
  • Look up .975 on any z table
  • The z value for .975 is 1.96
  • So, zc for a 95% CI is 1.96

HOW TO Get the Critical Value – Visual Tutorial

Now that you have zc, you can calculate E

E = (zc * s)/sqrt(n)

E = (1.96 * 2) / sqrt(100)

Note that “s” is the sample standard deviation. In our case, s = 2.

E = 3.92 / 10

E = .392


Now that you have E you can create/calculate the 95% CI


Remember, the 95% CI is:

sample mean – E < population mean < sample mean + E

65 – .392 < population mean < 65 + .392

64.61 < population mean < 65.39


In conclusion, the 95% CI for female height is {64.61, 65.39}. This tells us that 95% of the time we expect that the true population parameter (true mean female height) is between 64.61 and 65.39.

Example 2:

Some companies track health information to assist in determining the cost of life insurance policies. Yummy Coffee Company randomly sampled 100 recently paid policies and determined that the average age of clients in this sample is 80 years with a standard deviation of 3.6. What is the 90% confidence interval for the true mean age of its life insurance policy holders?


Here, we are looking for a 90% CI for a true population mean.

We are told that:

sample mean  = 80

sample std dev = 3.6

sample size n = 100


A CI always looks like:

sample mean – E    <   population mean  <  sample mean +E

So, to get the CI range, we need “E” the error.

In this case, we want the mean:

E = (zc * s)/sqrt(n)

zc is the critical value from the z table for the 2-tailed CI of 90%.

How to Get zc:

A 90% CI is .90.

1 – .90 = .10 in BOTH tails.

Using a cumulative z table, we need just the right side:

.10/2 = .05 (in the right tail)

1 – .05 = .95 (the cumulative area)

If you look up .9500 on any z table, you will get a z value of 1.645.

This tells us that zc is 1.645

So, now we can use the formula to find E

E = (zc * s)/sqrt(n)

E = (1.645 * 3.6)/sqrt(100) = (1.645*3.6) / 10 = 5.922/10 = .5922


A CI always looks like:

sample mean – E    <   population mean   <  sample mean +E

80 – .5922 < population mean < 80 + .5922

79.4 < population mean < 80.6

Conclusion: The 90% confidence interval is {79.4, 80.6}. We know that 90% of the time we expect the true population mean for the age to be between these two values.