Statistics6.1: Confidence Intervals for the Mean (large samples)
Objective 1: I can find a point estimate and calculate a margin of error.
In this chapter, we will use sample statistics to estimate the value of an unknown population parameter. Particularly, in section 6.1, we will learn how to estimate the ______parameter _____ when the sample size is ______or when the population is ______
______and ____ is known.
A ______is a single value estimate for a ______parameter. Every population parameter has a point estimate.
* The point estimate for _____ is _____.
* The point estimate for _____ or _____ is _____ or _____
* The point estimate for _____ is _____. (We have not seen this one yet in stats)
TIY 1: Another random sample of the number of sentences found in 30 magazine advertisements is listed below. Use this sample to find another point estimate for .
In both examples above, the probability that the population mean is exactly 12.4 or 14.8 is virtually zero. So, instead of estimating to be an exact number, it is often more useful to find an interval that we are pretty certain lies in. This is called an ______.
An ______is an interval, or range of values, used to estimate a population parameter.
Although you can assume that the point estimate in Ex 1 is not equal to the actual population mean, it is probably close to it. To form an interval estimate, use the point estimate as the center of the interval, then add and subtract a ______. For instance, if the margin of error is 2.1 (which means our estimated mean could be wrong by 2.1), then an interval estimate would be:
Before we are able to calculate a margin of error, we first must know how confident we need to be that our interval estimate contains the ______.
The ______c, is the probability that the interval contains the population parameter. If c = 0.90, then 90/100 times we perform this experiment, the mean DOES fall within the range of values of our confidence interval.
The level of confidence, c, is the area under the standard normal curve between -z and z.
-z and z are also called the ______. Let’s look at how to find critical values for some different c-levels, or ______levels.
If c = 90% or c = 0.90, the critical values (the values of that contain 90% of the area under the standard normal curve) can be found by the following steps:
1)
2)
3)
4)
5)
These steps should look familiar from section 5.3.
These are the most common levels of confidence used in this chapter, along with their critical values. This can be found on the bottom of the -z standard normal table in your book. Other values of c you will have to do by hand.
Now that you know how to find the critical values for levels of confidence, let’s look at how to calculate a ______. Given a level of confidence, ____, the margin of error, ____, is found using the formula
In order to use this formula, it is assumed that _____ is known. This is rarely the case, but as long as ______, ____ can be used in place of _____.
TIY 2: In TIY 1, we used a sample of 30 magazines to find the sample mean number of sentences per ad; it was = ____. Use a 90% level of confidence to find the margin of error for the mean number of sentences per magazine ad.
Objective 2: I can create confidence intervals for the population mean.
Using a point estimate and a margin or error, you can now construct an ______for a population parameter, such as . This interval estimate is called a ______
______.
Steps to creating a confidence interval by hand:
1) Find the sample statistics / 1)2) Specify _____, if known. Otherwise, if , use ____ in place of _____. / 2)
3) Find the critical value, ____, that corresponds to the given level of confidence. / 3)
4) Find the margin of error, E. / 4)
5) Find the left and right ______and construct the ______. / 5)
There are 3 ways to write a confidence interval (CI).
1)
2)
3)
TIY 3: Use the data in TIY 1 to construct a 95% CI for the mean number of sentences in all magazine ads. Compare this with the interval from Example 3.
Example 4: A college admissions director wishes to estimate the mean age of all students currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, the standard deviation is known to be 1.5 years and the population is normally distributed. Construct a 90% confidence interval of the population mean age.
Hey—wanna see the calculator shortcut??????
TIY 4: Redo Example 4 using the calculator.
Objective 3: I can determine the minimum sample size needed to estimate .
As c increases, the CI gets wider. As the interval widens, the precision of the estimate decreases. One way to improve the precision of our estimate without changing c is to increase n, the sample size. But how big should n be??
Given a confidence level, c, and a margin of error, E, the minimum sample size needed to estimate can be found by solving the margin of error formula for ______.
Recall that . Let’s resolve this formula for n.
Example 6: You want to estimate the mean number of sentences in a magazine ad. How many ads must be included in the sample of you want to be 95% confident that the sample mean is within one sentence of the population mean? Use the data from Ex 1.
TIY 6: How many magazine ads must be included in the sample if you want to be 95% confident that the sample mean is within two sentences of the population mean? Compare your answer with Example 6 above. Use the data from Ex 1.