1. 1 1 Slide © 2007 Thomson South-Western. All Rights Reserved Chapter 8 Interval Estimation Population Mean:  Known Population Mean:  Known Population Mean:  Unknown Population Mean:  Unknown n Determining the Sample Size n Population Proportion

2. 2 2 Slide © 2007 Thomson South-Western. All Rights Reserved A point estimator cannot be expected to provide the A point estimator cannot be expected to provide the exact value of the population parameter. exact value of the population parameter. A point estimator cannot be expected to provide the A point estimator cannot be expected to provide the exact value of the population parameter. exact value of the population parameter. An interval estimate can be computed by adding and An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. subtracting a margin of error to the point estimate. An interval estimate can be computed by adding and An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. subtracting a margin of error to the point estimate. Point Estimate +/  Margin of Error The purpose of an interval estimate is to provide The purpose of an interval estimate is to provide information about how close the point estimate is to information about how close the point estimate is to the value of the parameter. the value of the parameter. The purpose of an interval estimate is to provide The purpose of an interval estimate is to provide information about how close the point estimate is to information about how close the point estimate is to the value of the parameter. the value of the parameter. Margin of Error and the Interval Estimate

3. 3 3 Slide © 2007 Thomson South-Western. All Rights Reserved The general form of an interval estimate of a The general form of an interval estimate of a population mean is population mean is The general form of an interval estimate of a The general form of an interval estimate of a population mean is population mean is Margin of Error and the Interval Estimate

4. 4 4 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Mean:  Known n In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation , or the population standard deviation , or the sample standard deviation s the sample standard deviation s  is rarely known exactly, but often a good estimate can be obtained based on historical data or other information.  is rarely known exactly, but often a good estimate can be obtained based on historical data or other information. We refer to such cases as the  known case. We refer to such cases as the  known case.

5. 5 5 Slide © 2007 Thomson South-Western. All Rights Reserved There is a 1   probability that the value of a There is a 1   probability that the value of a sample mean will provide a margin of error of or less.    /2 1 -  of all values 1 -  of all values Sampling distribution of Sampling distribution of Interval Estimation of a Population Mean:  Known

6. 6 6 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimate of  Interval Estimate of  Interval Estimate of a Population Mean:  Known where: is the sample mean 1 -  is the confidence coefficient 1 -  is the confidence coefficient z  /2 is the z value providing an area of z  /2 is the z value providing an area of  /2 in the upper tail of the standard  /2 in the upper tail of the standard normal probability distribution normal probability distribution  is the population standard deviation  is the population standard deviation n is the sample size n is the sample size Margin of error

7. 7 7 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Mean: s Known n Example: A simple random sample of 64 items results in a sample mean of 25. The population standard deviation is 4. Provide a 95% confidence interval for the population mean. n Step 1: What is the value of α? 1 – α = 95%  α = 5% 1 – α = 95%  α = 5% n Step 2: Find the z α/2 value. z 0.05/2 = z 0.025 = 1.96 z 0.05/2 = z 0.025 = 1.96 n Step 3: Calculate the confidence interval 25 ± 1.96· 4/8 = 25 ± 0.98. 25 ± 1.96· 4/8 = 25 ± 0.98. In other words: In other words: Prob( 24.02 ≤ μ ≤ 25.98) = 95% Prob( 24.02 ≤ μ ≤ 25.98) = 95%

8. 8 8 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimate of a Population Mean:  Known n Adequate Sample Size In general, a sample size of n = 30 is adequate. In general, a sample size of n = 30 is adequate. If the population distribution is highly skewed or If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is contains outliers, a sample size of 50 or more is recommended. recommended. If the population distribution is highly skewed or If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is contains outliers, a sample size of 50 or more is recommended. recommended.

9. 9 9 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimate of a Population Mean:  Known n Adequate Sample Size (continued) If the population is believed to be at least If the population is believed to be at least approximately normal, a sample size of less than 15 approximately normal, a sample size of less than 15 can be used. can be used. If the population is believed to be at least If the population is believed to be at least approximately normal, a sample size of less than 15 approximately normal, a sample size of less than 15 can be used. can be used. If the population is not normally distributed but is If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice. will suffice. If the population is not normally distributed but is If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice. will suffice.

10. 10 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Mean:  Unknown If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . This is the  unknown case. This is the  unknown case. Main difference with the “σ known” case The interval estimate for  is based on the t distribution. Main difference with the “σ known” case The interval estimate for  is based on the t distribution.

11. 11 Slide © 2007 Thomson South-Western. All Rights Reserved t Distribution A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. Degrees of freedom refer to the number of Degrees of freedom refer to the number of independent pieces of information that go into the independent pieces of information that go into the computation of s. computation of s. Degrees of freedom refer to the number of Degrees of freedom refer to the number of independent pieces of information that go into the independent pieces of information that go into the computation of s. computation of s. The more degrees of freedom  The less dispersion. The more degrees of freedom  The less dispersion. The more degrees of freedom  The smaller the The more degrees of freedom  The smaller the difference between the t and the standard normal distribution.. The more degrees of freedom  The smaller the The more degrees of freedom  The smaller the difference between the t and the standard normal distribution..

12. 12 Slide © 2007 Thomson South-Western. All Rights Reserved t Distribution Standardnormaldistribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) of freedom) 0 z, t

13. 13 Slide © 2007 Thomson South-Western. All Rights Reserved For more than 100 degrees of freedom, the standard For more than 100 degrees of freedom, the standard normal z value provides a good approximation to normal z value provides a good approximation to the t value. the t value. For more than 100 degrees of freedom, the standard For more than 100 degrees of freedom, the standard normal z value provides a good approximation to normal z value provides a good approximation to the t value. the t value. t Distribution

14. 14 Slide © 2007 Thomson South-Western. All Rights Reserved n Interval Estimate where: 1 -  = the confidence coefficient t  /2 = the t value providing an area of  /2 t  /2 = the t value providing an area of  /2 in the upper tail of a t distribution in the upper tail of a t distribution with n - 1 degrees of freedom with n - 1 degrees of freedom s = the sample standard deviation s = the sample standard deviation Interval Estimation of a Population Mean:  Unknown Margin of error

15. 15 Slide © 2007 Thomson South-Western. All Rights Reserved Interval Estimation of a Population Mean: s Unknown n Example: A simple random sample of 64 items results in a sample mean of 25. The sample standard deviation is 4. Provide a 95% confidence interval for the population mean. n Step 1: What is the value of α? 1 – α = 95%  α = 5% 1 – α = 95%  α = 5% n Step 2: Find the t α/2 value. Use n-1 = 64 – 1 = 63 degrees of freedom t 0.05/2 = t 0.025 = 1.998 t 0.05/2 = t 0.025 = 1.998 n Step 3: Calculate the confidence interval 25 ± 1.998· 4/8 = 25 ± 0.999. 25 ± 1.998· 4/8 = 25 ± 0.999. In other words: In other words: Prob( 24.001 ≤ μ ≤ 25.999) = 95% Prob( 24.001 ≤ μ ≤ 25.999) = 95%

16. 16 Slide © 2007 Thomson South-Western. All Rights Reserved Summary of Interval Estimation Procedures for a Population Mean Can the population standard deviation  be assumed deviation  be assumed known ? known ? Use the sample standard deviation s to estimate s Use YesNo Use  Known Case  Unknown Case

17. 17 Slide © 2007 Thomson South-Western. All Rights Reserved Let E = the desired margin of error. Let E = the desired margin of error. How can we find the sample size (n) needed to achieve How can we find the sample size (n) needed to achieve a specific margin of error (E)? How can we find the sample size (n) needed to achieve How can we find the sample size (n) needed to achieve a specific margin of error (E)? Sample Size for an Interval Estimate of a Population Mean Solve for n The above methodology can be used even if σ is not known. The above methodology can be used even if σ is not known. In this case, a planning value of σ is needed. The above methodology can be used even if σ is not known. The above methodology can be used even if σ is not known. In this case, a planning value of σ is needed.

18. 18 Slide © 2007 Thomson South-Western. All Rights Reserved The general form of an interval estimate of a The general form of an interval estimate of a population proportion is population proportion is The general form of an interval estimate of a The general form of an interval estimate of a population proportion is population proportion is Interval Estimation of a Population Proportion Key for the computation of Margin of Error: Key for the computation of Margin of Error: The sampling distribution of The sampling distribution of Key for the computation of Margin of Error: Key for the computation of Margin of Error: The sampling distribution of The sampling distribution of The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever np > 5 and by a normal distribution whenever np > 5 and n (1 – p ) > 5. n (1 – p ) > 5. The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever np > 5 and by a normal distribution whenever np > 5 and n (1 – p ) > 5. n (1 – p ) > 5.

19. 19 Slide © 2007 Thomson South-Western. All Rights Reserved  /2 Interval Estimation of a Population Proportion n Normal Approximation of Sampling Distribution of Sampling distribution of Sampling distribution of p p 1 -  of all values 1 -  of all values

20. 20 Slide © 2007 Thomson South-Western. All Rights Reserved n Interval Estimate Interval Estimation of a Population Proportion where: 1 -  is the confidence coefficient z  /2 is the z value providing an area of z  /2 is the z value providing an area of  /2 in the upper tail of the standard  /2 in the upper tail of the standard normal probability distribution normal probability distribution is the sample proportion is the sample proportion

21. 21 Slide © 2007 Thomson South-Western. All Rights Reserved Solving for the necessary sample size, we get n Margin of Error Sample Size for an Interval Estimate of a Population Proportion However, will not be known until after we have selected the sample. We will use the planning value p * for.

22. 22 Slide © 2007 Thomson South-Western. All Rights Reserved Sample Size for an Interval Estimate of a Population Proportion The planning value p * can be chosen by: 1. Using the sample proportion from a previous sample of the same or similar units, or 2. Selecting a preliminary sample and using the sample proportion from this sample. n Necessary Sample Size