1. 6.1 Confidence Intervals for the Mean (Large Samples)
Statistics
Spring Semester
Mrs. Spitz

2. Objectives/Assignment
How to find a point estimate and a maximum error of estimate
How to construct and interpret confidence intervals for the population mean
How to determine the required minimum sample size when estimating 
Assignment: pp #1-40 all

3. Estimating Population Parameters
In this chapter, you will learn an important technique of statistical inference—to use sample statistics to estimate the value of an unknown population parameter. In this particular section, you will learn how to use sample data to make an estimate of the population parameter  when the sample size is at least 30 or the standard deviation  is known. To make such an inference, begin by fnding a point estimate.

4. DEFINITION
A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean  is the sample mean .

5. Ex. 1: Finding a Point Estimate
Market researchers use the number of sentences per advertisement as a measure of readability for magazine advertisements. The following represents a random sample of the number of sentences found in 54 advertisements. Find a point estimate of the population mean .
So, your point estimate for the mean length of all magazine advertisements is 12.4 sentences.

6. You try it: Finding a Point Estimate
Another random sample of the number of sentences found in 30 magazine advertisements is listed below. Use this sample to find another point estimate of the population mean .
Find the sample mean.
Estimate the mean sentence length of the population.
Number of Sentences 16 9 14 11 17 12 99 18 13 5 6 20 7 4

7. More about point estimates
In Example 1, the probability that the population mean is exactly 12.4 is virtually zero. So, instead of estimating  to be exactly 12.4, you can increase your accuracy by estimating that it lies within an interval. This is called “making an interval estimate.”

8. More about interval estimates
Although you can assume that the point estimate in Example 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 margin of error. For instance, if the margin of error is 2.1, then an interval estimate would be given by 12.4 ± 2.1 or 10.3 <  < The point estimate and interval estimate are as follows.
Before finding an interval estimate, you should first determine how confident you need to be that your interval contains the population mean .

9. Study Tip:
In this course, you will usually use 90%, 95%, and 99% levels of confidence. The following z-scores correspond to these levels of confidence.
Level of Confidence zc
90%
1.645
95%
1.96
99%
2.575

10. DEFINITION
The level of confidence, c, is the probability that the interval estimate contains the population parameter.

11. Critical Values
You know from the Central Limit Theorem that when n ≥ 30, the sampling distribution of sample means is a normal distribution. The level of confidence, c, is the area under the standard normal curve between the critical values, -zc and zc.

12. Critical Values
You can see from the graph that c is the percent of area under the normal curve between -zc and zc. The area remaining is 1 – c, so the area in each tail is ½ (1 – c). For instance, if c = 90%, then 5% of the area lies to the left of -zc = and 5% lies to the right of zc = 1.645

13. Error of Estimate
The distance between the point estimate and the actual parameter value is called the error of estimate. When estimating , the error of estimate is the distance
In most cases, of course,  is unknown and varies from sample to sample. However, you an calculate a maximum value for the error if you know the level of confidence and the sampling distribution.

14. DEFINITION
Given a level of confidence, the maximum error of estimate (or error tolerance), E, is the greatest possible distance between the point estimate and the value of the parameter it is estimating.
When n ≥ 30, the sample standard deviation s can be used in place of .

15. Ex. 2: Finding the Maximum Error of Estimate
Use the data in Example 1 and a 95% confidence interval to find the maximum error of estimate for the number of sentences in a magazine advertisement.

16. Ex. 2: Finding the Maximum Error of Estimate--Solution
The z-score that corresponds to a 95% confidence interval is This implies that 95% of the area under the curve falls within 1.96 standard deviations of the mean. You don’t know the populations standard deviation . But because n ≥ 30, you can use s in place of .

17. Ex. 2: Finding the Maximum Error of Estimate--Solution
Using the values zc = 1.96, s ≈5.0, and n = 54.
You are 95% confident that the maximum error of estimate is about 1.3 sentences per magazine advertisement.

18. Confidence Intervals for the Population Mean
Using a point estimate and a maximum error of estimate, you can construct an interval estimate of a population parameter such as . This interval estimate is called a confidence interval.

20. Ex. 3: Constructing a Confidence Interval
Construct a 95% confidence interval for the mean number of sentences in a magazine advertisement.
Solution: In examples 1 and 2, you found that
= 12.4 and E = The confidence inteval is as follows:

21. Insight:
A larger sample size tends to give you a narrower confidence interval—for the same level of confidence.

22. Ex. 4: Constructing a Confidence Interval using Technology
Use a graphing calculator to construct a 99% confidence interval for the mean number of sentences in a magazine advertisement, using the sample in Example 1.
Solution: To use a calculator to solve the problem, enter the data and find that the sample standard deviation is s≈ Then, use the confidence interval command to calculate the confidence interval (ZInterval for the TI-83 or TI-84). The display should look like the one on the next slide.

23.
So, a 99% confidence interval for  is (10.7, 14.2). With 99% confidence, you can say that the mean number of sentences is between 10.7 and 14.2.

24. More about intervals
In Ex. 4, and Try it Yourself 4, the same sample data was used to construct intervals with different levels of confidence. Notice that as the level of confidence increases, the width of the confidence interval also increases. In other words, using the same sample data, the greater the level of confidence, the wider the interval.

25. In Ex. 5, notice that if  is known, then the sample size can be less than 30.
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. Construct a 90% confidence interval of the population mean age.

26. The 90% confidence interval is as follows:
In Ex. 5, notice that if  is known, then the sample size can be less than 30.
The 90% confidence interval is as follows:
So, with 90% confidence, you can say that the mean age of the students is between and years.

27. Interpreting the results
After constructing a confidence interval, it is important that you interpret the results correctly. Consider the 90% confidence interval constructed in ex. 5. Because  already exists, it is either in the interval or not. It is not correct to say, “There is a 90% probability that the actual mean is in the interval (22.51, ”
The correct way to interpret your confidence interval is, “There is a 90% probability that the confidence interval you described contains .” This means also, of course, that there is a 10% probability that your confidence interval WILL NOT contain .

28. Sample Size
As the level of confidence increases, the confidence interval widens. As the confidence interval widens, the precision of the estimate decreases. One way to improve the precision of an estimate without decreasing the level of confidence is to increase the sample size. But how large a sample size is needed to guarantee a certain level of confidence for a given maximum error of estimate?

29. Ex. 6: Determining a Minimum Sample Size
You want to estimate a mean number of sentences in a magazine advertisement. How many magazine advertisements must be included in the sample if you want to be 95% confident that the sample mean is within one sentence of the population mean?

30. Ex. 6: Determining a Minimum Sample Size
Using c = 0.95, zc = 1.96, s ≈5.0 (from example 2), and E = 1, you can solve for the minimum sample size, n.
When necessary, round up to obtain a whole number. So, you should include at least 97 magazine advertisements in your sample. (You already have 54, so you need 43 more.)