1. Estimating Population Values
Chapter 7
Estimating Population Values

2. Chapter 7 - Chapter Outcomes
After studying the material in this chapter, you should be able to:
Distinguish between a point estimate and a confidence interval estimate.
Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions.

3. Chapter 7 - Chapter Outcomes (continued)
After studying the material in this chapter, you should be able to:
Determine the required sample size for an estimation application involving a single population mean.
Establish and interpret a confidence interval estimate for a single population proportion.

4. Point Estimates
A point estimate is a single number determined from a sample that is used to estimate the corresponding population parameter.

5. Sampling Error
Sampling error refers to the difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population.

6. Confidence Intervals
A confidence interval refers to an interval developed from randomly sample values such that if all possible intervals of a given width were constructed, a percentage of these intervals, known as the confidence level, would include the true population parameter.

7. Lower Confidence Limit Upper Confidence Limit
Confidence Intervals
Lower Confidence Limit
Upper Confidence Limit
Point Estimate

8. 95% Confidence Intervals (Figure 7-3)
0.95
z.025= -1.96
z.025= 1.96

9. Confidence Interval - General Format -
Point Estimate  (Critical Value)(Standard Error)

10. Confidence Intervals
The confidence level refers to a percentage greater than 50 and less than 100 that corresponds to the percentage of all possible confidence intervals, based on a given size sample, that will contain the true population value.

11. Confidence Intervals
The confidence coefficient refers to the confidence level divided by 100% -- i.e., the decimal equivalent of a confidence level.

12. Confidence Interval - General Format:  known -
Point Estimate  z (Standard Error)

13. Confidence Interval Estimates
CONFIDENCE INTERVAL ESTIMATE FOR  ( KNOWN)
where:
z = Critical value from standard normal table
 = Population standard deviation
n = Sample size

14. Example of a Confidence Interval Estimate for 
A random sample of 100 cans, from a population with  = 0.20, produced a sample mean equal to A 95% confidence interval would be:
ounces
ounces

15. Special Message about Interpreting Confidence Intervals
Once a confidence interval has been constructed, it will either contain the population mean or it will not. For a 95% confidence interval, if you were to produce all the possible confidence intervals using each possible sample mean from the population, 95% of these intervals would contain the population mean.

16. Margin of Error
The margin of error is the largest possible sampling error at the specified level of confidence.

17. MARGIN OF ERROR (ESTIMATE FOR  WITH  KNOWN)
where:
e = Margin of error
z = Critical value
= Standard error of the sampling distribution

18. Example of Impact of Sample Size on Confidence Intervals
If instead of random sample of 100 cans, suppose a random sample of 400 cans, from a population with  = 0.20, produced a sample mean equal to A 95% confidence interval would be:
ounces
ounces
n=400
n=100
ounces
ounces

19. Student’s t-Distribution
The t-distribution is a family of distributions that is bell-shaped and symmetric like the standard normal distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.

20. Degrees of freedom
Degrees of freedom refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - k.

21. t-Values t-VALUE where: = Sample mean = Population mean
s = Sample standard deviation
n = Sample size

22. Confidence Interval Estimates
( UNKNOWN)
where:
t = Critical value from t-distribution with n-1 degrees of freedom
= Sample mean
s = Sample standard deviation
n = Sample size

23. Confidence Interval Estimates
CONFIDENCE INTERVAL-LARGE SAMPLE WITH  UNKNOWN
where:
z =Value from the standard normal distribution
= Sample mean
s = Sample standard deviation
n = Sample size

24. Determining the Appropriate Sample Size
SAMPLE SIZE REQUIREMENT - ESTIMATING  WITH  KNOWN
where:
z = Critical value for the specified confidence interval
e = Desired margin of error
 = Population standard deviation

25. Pilot Samples
A pilot sample is a random sample taken from the population of interest of a size smaller than the anticipated sample size that is used to provide and estimate for the population standard deviation.

26. Example of Determining Required Sample Size (Example 7-7)
The manager of the Georgia Timber Mill wishes to construct a 90% confidence interval with a margin of error of 0.50 inches in estimating the mean diameter of logs. A pilot sample of 100 logs yields a sample standard deviation of 4.8 inches.
Note, the manager needs only 150 more logs since the 100 in the pilot sample can be used.

27. Estimating A Population Proportion
SAMPLE PROPORTION
where:
x = Number of occurrences
n = Sample size

28. Estimating a Population Proportion
STANDARD ERROR FOR p
where:
 =Population proportion
n = Sample size

29. Confidence Interval Estimates for Proportions
CONFIDENCE INTERVAL FOR 
where:
p = Sample proportion
n = Sample size
z = Critical value from the standard normal distribution

30. Example of Confidence Interval for Proportion (Example 7-8)
62 out of a sample of 100 individuals who were surveyed by Quick-Lube returned within one month to have their oil changed. To find a 90% confidence interval for the true proportion of customers who actually returned:
0.54
0.70

31. Determining the Required Sample Size
MARGIN OF ERROR FOR ESTIMATING 
where:
 = Population proportion
z = Critical value from standard normal distribution
n = Sample size

32. Determining the Required Sample Size
SAMPLE SIZE FOR ESTIMATING 
where:
 = Value used to represent the population proportion
e = Desired margin of error
z = Critical value from the standard normal table

33. Key Terms Confidence Coefficient Confidence Interval Confidence Level
Degrees of Freedom
Margin of Error
Pilot Sample
Point Estimate
Sampling Error
Student’s t-distribution