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In section 7.2 we looked at constructing confidence intervals for a population proportion. In
7.3 and 7.4 we're going to construct confidence intervals for a population mean. The nice
thing is that it's almost the same as before- find a point estimate, find the margin of
error and construct the interval. The formulas change a little bit, but the process is the
same. First we are going to construct confidence
intervals for the mean when we know what the population standard deviation is. To do this,
we need a simple random sample, we need to know what sigma is, and we need to meet the
criteria to apply the central limit theorem, so the population needs to be normally distributed
or the sample size needs to be greater than 30. If we've met those criteria, we can create
a confidence interval for mu by using the sample mean, x-bar, as the best point estimate,
and an error equal to z alpha over two times sigma over the square root of n.
This example says that a survey of the weights of 40 men has a mean of 172.55 pounds and
that other research shows that the population standard deviation of weights of men is 26
pounds. We are going to construct a 99% confidence interval for the population mean weight of
all men by first finding the best point estimate for mu. The sample mean is the best estimate
of the population mean and our sample had a mean of 172.55 pounds, so x-bar equals 172.55
is the best point estimate of mu. Next we need to calculate the margin of error,
so we need to know z alpha over two, sigma and n. At 99% confidence, alpha is 0.01, so
alpha over two is 0.005 and the inverse normal function tells us that z 0.005 is 2.58. The
statement of the problem mentions two different standard deviations. We need sigma, which
is the population standard deviation, so the value of sigma is 26. Last we need the sample
size, n, which is 40. We put those into the calculator and get that
E is equal to 10.606. Our interval for mu is x bar minus E to x bar plus E. x-bar is
172.55 and E is 10.606, so that comes out to 161.944 on the bottom and 183.156 at the
top. We are 99% confident that that interval holds the true mean weight of the population
of all men. The safety board works under the assumption
that the mean weight of men is 166.3 pounds. Does this seem valid based on our results?
Well, our interval goes from 161.944 pounds to 183.156 pounds, and the safety board's
weight of 166.3 pounds is in that range, so it's plausible. We don't have reason to reject
it and say it's wrong the way we could if it fell outside of the confidence interval.
Next up, just like before, we can rearrange the formula for the margin of error to get
a formula for the sample size n that is needed to get the desired bound on the error. Also
as before, we will always round decimal values up with these computations.
So let's suppose we want to estimate the mean IQ score for a population of students and
we want 95% confidence that the sample mean is within 3 IQ points of the population mean.
How many students do we need to survey if the population standard deviation is 15 IQ
points? Looking at the formula for n, we see that
we need to find the critical value corresponding to 95% confidence. The inverse normal function
gives us a value of 1.96, sigma is 15 and E is 3. When we did this type of problem in
the last section, our bound on the error was 3 percentage points, so we had to write 0.03
for E in the formula. Here, this is 3 points, not 3 percentage points, so it stays as 3.
We plug them in and get 96.04 which gets rounded to 97. In order to be 95% confident that the
sample mean is within 3 percentage points of the population mean the minimum sample
size is 97 students.