Welcome! In this video, I’ll be constructing confidence
intervals for a population mean, assuming the population standard deviation is known.
A confidence interval, constructed from sample data, is a range of values that is likely
to include the population parameter, at some specified confidence level.
The confidence interval for a population mean is determined by taking the sample mean (the
point estimate) and subtracting and adding a margin of error to it.
So if the population standard deviation is known, the margin of error is determined by
z_α⁄2 × σ/√n
where α, the significance level, is 1 minus the confidence level.
And correspondingly, the confidence level is 1- α.
So if the confidence level is 95%, α will be 1 – 0.95 which equals 0.05.
z_α⁄2 here is a single value, called the critical value.
It can be found in the normal tables or by using software.
When constructing a 95% confidence interval, for example,
the confidence level (0.95) is in the middle of the distribution,
and the remaining 0.05 (alpha) is divided equally into the 2 tails as 0.025 each.
From the less-than cumulative standard normal tables, 0.025 in the left tail corresponds
to a z value of negative 1.96. And, due to symmetry, it will be positive
1.96 for the right tail. So the z critical value corresponding to the
95% confidence level (which we write as z.025) is 1.96.
Let’s look at an example: Scores on an exam are normally distributed
with a population standard deviation of 5.6. A random sample of 40 scores on the exam has
a mean of 32. We want to construct confidence interval estimates
for the population mean at 80, 90, and 98% confidence levels.
The margin of error is going to be zα/2 times 5.6 divided by the square root of 40.
We now need to find the critical value for each confidence level.
For the 80% confidence interval, alpha is 0.2, so we have 0.1 in each tail.
Looking up 0.1 in the normal tables, we find the closest value to be 0.1003. And that corresponds
to a z-score of -1.28 in the lower tail. And because of symmetry, it is 1.28 for the
upper tail. So the z-critical value for the 80% confidence
level is 1.28. The margin of error is therefore 1.13.
So the lower limit of the confidence interval is 32 – 1.13 which gives 30.87.
And the upper limit is 32 + 1.13 which gives 33.13.
To interpret that we say, we are 80% confident that the population mean score is between
30.87 and 33.13. With 90% confidence, alpha is 0.1, so we have
0.05 in each tail. Looking that up in the z- tables we find that
0.05 lies exactly midway between 0.0505 and 0.0495 here. That is, between z-scores of
-1.64 and -1.65. So we average the 2 values to obtain -1.645.
And so the critical value is 1.645. And that gives a margin of error of 1.46.
The lower limit of the 90% confidence interval is thus 30.54 and the upper limit is 33.46.
The 98% confidence interval has 0.01 in each tail and has a z-critical value of 2.33.
So the margin of error will be 2.06. The lower limit will be 29.94 and the upper
limit 34.06.
From the results, we see that, as confidence level increases,
the critical value z increases, the margin of error increases,
and consequently, the confidence interval became wider.
And that’s it for this video. Thanks for watching.