1. Elementary Statistics
Confidence Intervals 6
Elementary Statistics
Larson Farber

2. Confidence Intervals for the Mean
Section 6.1
Confidence Intervals for the Mean
(large samples)

3. Point Estimate
A point estimate is a single value estimate for a population parameter.
The best point estimate of the population mean is the
sample mean
An unbiased estimator is just as likely to overestimate as to underestimate the parameter it is estimating. The sample mean is an unbiased estimator of the population mean. From chapter 5 students know that the mean of the sample means is the population mean.

4. Example: Point Estimate
A random sample of 35 airfare prices (in dollars) for a one-way ticket from Atlanta to Chicago. Find a point estimate for the population mean . 99
101
107
102
109 98
105
103
101
105 98
107
104 96
105 95 98 94
100
104
111
114 87
104
108
101 87
103
106
117 94
103
101
105 90
Sometimes students will work from raw data and other times summary statistics. The raw data are presented here but summary statistics are given in the interest of time. After raw data is given, ask students what single number could be used to estimate the population mean.

5. Example: Point Estimate
A random sample of 35 airfare prices (in dollars) for a one-way ticket from Atlanta to Chicago. Find a point estimate for the population mean . 99
101
107
102
109 98
105
103
101
105 98
107
104 96
105 95 98 94
100
104
111
114 87
104
108
101 87
103
106
117 94
103
101
105 90
The sample mean is
Sometimes students will work from raw data and other times summary statistics. The raw data are presented here but summary statistics are given in the interest of time. After raw data is given, ask students what single number could be used to estimate the population mean.
The point estimate for the price of all one way tickets from Atlanta to Chicago is $

6. Interval Estimates • • Point estimate
101.77
An interval estimate is an interval or range of values used to estimate a population parameter. ( ) •
101.77
The chances the the archer hits the exact point center are virtually 0. The probability he hits within the center ring is his level of confidence for that ring. Discuss as the rings widen, the level of confidence goes up but accuracy is sacrificed.
The level of confidence c is the probability that the interval estimate contains the population parameter.

7. Distribution of Sample Means
When the sample size is at least 30, the sampling distribution for is normal.
Sampling distribution of
For c = 0.95
0.95
Have students calculate other critical numbers for other values of c.
For c = 90%, z.90= 1.645
For c = 99%, z.99 =1.575
0.025
0.025 z
-1.96
1.96
95% of all sample means will have standard scores between z = and z = 1.96

8. Distribution of Sample Means
For c = 90%, z.90= 1.645
For c = 95%, z.95= 1.96
For c = 99%, z.99 =2.575
Have students calculate other critical numbers for other values of c.
For c = 90%, z.90= 1.645
For c = 99%, z.99 =1.575

9. Sampling Error
The difference between the point estimate and the actual parameter mean value is called the sampling error.
Have students calculate other critical numbers for other values of c.
For c = 90%, z.90= 1.645
For c = 99%, z.99 =1.575

10. Maximum Error of Estimate
The margin of error E is the greatest possible distance between the point estimate and the value of the parameter it is estimating for a given level of confidence c.
Calculate the error of estimate for c = .90 and c= .95. Have students compare the widths of the intervals at various levels of confidence.
When n > 30, the sample standard deviation s can be used in place of σ

11. Maximum Error of Estimate
Find E, the maximum error of estimate for the one-way plane fare from Atlanta to Chicago for a 95% level of confidence given s = 6.69.
Calculate the error of estimate for c = .90 and c= .95. Have students compare the widths of the intervals at various levels of confidence.

12. Maximum Error of Estimate
Find E, the maximum error of estimate for the one-way plane fare from Atlanta to Chicago for a 95% level of confidence given s = 6.69.
Using zc = 1.96, s = 6.69, and n = 35,
Calculate the error of estimate for c = .90 and c= .95. Have students compare the widths of the intervals at various levels of confidence.
You are 95% confident that the maximum error of estimate is $2.22.

13. Confidence Intervals for
A confidence interval for the population mean is
Find the 95% confidence interval for the one-way plane fare from Atlanta to Chicago.
Encourage students to use a number line to form confidence intervals.

14. Confidence Intervals for
We found = and E = 2.22
Left endpoint
Right endpoint •
101.77 ( )
Encourage students to use a number line to form confidence intervals.
103.99
99.55
With 95% confidence, you can say the mean one-way fare from Atlanta to Chicago is between $99.55 and $

15. Sample Size
Given a confidence level and a margin of error, the minimum sample size n needed to estimate the population mean is
Rework the problem with different values of E. Compare. What effect does increasing E, the maximum tolerance have on the minimum required sample?

16. Sample Size
You want to estimate the mean one-way fare from Atlanta to Chicago. How many fares must be included in your sample if you want to be 95% confident that the sample mean is within $2 of the population mean?
Rework the problem with different values of E. Compare. What effect does increasing E, the maximum tolerance have on the minimum required sample?

17. Sample Size
You want to estimate the mean one-way fare from Atlanta to Chicago. How many fares must be included in your sample if you want to be 95% confident that the sample mean is within $2 of the population mean?
Rework the problem with different values of E. Compare. What effect does increasing E, the maximum tolerance have on the minimum required sample?
You should include at least 43 fares in your sample. Since you already have 35, you need 8 more.

18. Confidence Intervals for the Mean
Section 6.2
Confidence Intervals for the Mean
(small samples)

19. The t-Distribution
If the distribution of a random variable x is normal and n < 30, then the sampling distribution of is a t-distribution with n – 1 degrees of freedom.
Sampling distribution
n = 13
d.f. = 12
c = 90%
.90
.05
.05
Discuss degrees of freedom. Find other critical numbers for different sample sizes. Show other t-distributions. If the number of degrees of freedom is 30 or more, the t-distributions are very close to the standard normal distribution. t
-1.782
1.782
The critical value for t is % of the sample means (n = 13) will lie between t = and t =

20. Confidence Interval–Small Sample
Maximum error of estimate
In a random sample of 13 American adults, the mean waste recycled per person per day was 4.3 pounds and the standard deviation was 0.3 pound. Assume the variable is normally distributed and construct a 90% confidence interval for .
1. The point estimate is = 4.3 pounds
The process for forming a confidence interval is virtually the same as that for large samples. Note though that the population standard deviation cannot be estimated by s.
2. The maximum error of estimate is

21. Confidence Interval–Small Sample
1. The point estimate is = 4.3 pounds
2. The maximum error of estimate is
Right endpoint
Left endpoint
The process for forming a confidence interval is virtually the same as that for large samples. Note though that the population standard deviation cannot be estimated by s. •
4.3 ( )
4.152
4.448
4.15 < < 4.45
With 90% confidence, you can say the mean waste recycled per person per day is between 4.15 and 4.45 pounds.

22. Confidence Intervals for Population Proportions
Section 6.3
Confidence Intervals for Population Proportions

23. Confidence Intervals for Population Proportions
The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample
(Read as p-hat)
is the point estimate for the proportion of failures where
Remind students that when they used a normal approximation to the binomial the conditions of np >5 and nq>5 had to be met. In estimating the standard error use the point estimates.
If and the sampling distribution for is normal.

24. Confidence Intervals for Population Proportions
The maximum error of estimate, E, for a x-confidence interval is:
A c-confidence interval for the population proportion, p, is
Remind students that when they used a normal approximation to the binomial the conditions of np >5 and nq>5 had to be met. In estimating the standard error use the point estimates.

25. Confidence Interval for p
In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related.
The technique used is similar to the technique used to form confidence intervals for the population mean. Start with the point estimate at the center and add and subtract the maximum error of estimate to form the endpoints of the confidence interval.

26. Confidence Interval for p
In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related.
1. The point estimate for p is
(.235) > 5 and 1907(.765) > 5, so the sampling distribution is normal.
The technique used is similar to the technique used to form confidence intervals for the population mean. Start with the point estimate at the center and add and subtract the maximum error of estimate to form the endpoints of the confidence interval. 3.

27. Confidence Interval for p
In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related.
Left endpoint
Right endpoint ) •
.235 (
The technique used is similar to the technique used to form confidence intervals for the population mean. Start with the point estimate at the center and add and subtract the maximum error of estimate to form the endpoints of the confidence interval.
.21
.26
0.21 < p < 0.26
With 99% confidence, you can say the proportion of fatal accidents that are alcohol related is between 21% and 26%.

28. Minimum Sample Size
If you have a preliminary estimate for p and q, the minimum sample size given a x-confidence interval and a maximum error of estimate needed to estimate p is:
If you do not have a preliminary estimate, use 0.5 for both
The maximum value of pq occurs when p is 0.5 and q=0.5. Show students several examples of p, have them calculate q and then pq. For students who have had calculus, f(p)=p(1-p) =p-p2 will be a maximum when 1-2p=0. Algebra students can see that the graph of f(p) is a parabola and its vertex is at p=.05, q = .05.

29. Example–Minimum Sample Size
You wish to estimate the proportion of fatal accidents that are alcohol related at a 99% level of confidence. Find the minimum sample size needed to be be accurate to within 2% of the population proportion.
With no preliminary estimate use 0.5 for
Discuss the advantage of having a preliminary estimate.
You will need at least 4415 for your sample.

30. Example–Minimum Sample Size
You wish to estimate the proportion of fatal accidents that are alcohol related at a 99% level of confidence. Find the minimum sample size needed to be be accurate to within 2% of the population proportion. Use a preliminary estimate of p =
Discuss the advantage of having a preliminary estimate.
With a preliminary sample you need at least
n = 2981 for your sample.

31. Confidence Intervals for Variance and Standard Deviation
Section 6.4
Confidence Intervals for Variance and Standard Deviation

32. The Chi-Square Distribution
The point estimate for is s2 and the point estimate for is s.
If the sample size is n, use a chi-square x2 distribution with n – 1 d.f. to form a c-confidence interval.
.95
6.908
28.845
Find R2 the right-tail critical value and xL2 the left-tail
critical value for c = 95% and n = 17.
Unlike the z and t distribution, chi-square is always positive and is not a symmetric curve. (It is skewed right.)
When the sample size is 17, there are 16 d.f.
Area to the right of xR2 is (1 – 0.95)/2 = and area to the right of xL2 is ( )/2 = 0.975
xL2 = 6.908
xR2 =

33. Confidence Intervals for
A c-confidence interval for a
population variance is:
To estimate the standard deviation take the square root of each endpoint.
You randomly select the prices of 17 CD players. The sample standard deviation is $150. Construct a 95% confidence interval for and .

34. Confidence Intervals for
To estimate the standard deviation take the square root of each endpoint.
Find the square root of each part.
You can say with 95% confidence that is
between and
and between $ and $