1. 1 Chapter 8 Interval Estimation

2. 2 Chapter Outline  Population Mean: Known  Population Mean: Unknown  Population Proportion

3. 3 Introduction  The sampling distribution introduced last chapter connects sample statistics to population parameters.  In reality, we probably don’t know any of the population parameters. However, a study on sampling distributions can provide reasonable references to the population parameters.

4. 4 Margin of Error and the Interval Estimate   A point estimator cannot be expected to provide the exact value of the population parameter. For instance, the probability of any particular sample mean equal population mean is zero, i.e. p ( =    An interval estimate can be computed by adding and subtracting a margin of error to the point estimate.  The purpose of an interval estimate is to provide a reasonable value range of the population parameters.

5. 5 Margin of Error and the Interval Estimate  The general form of an interval estimates of a population mean  is

6. 6 Interval Estimate of A Population Mean:  Known  In the first scenario, we assume  to be known. Although  is rarely known in reality, a good estimate can be obtained based on historical data or other information.  Let’s use the example of Checking Accounts from last chapter as an illustration. Here, we assume that the population standard deviation is known (  =66). Our goal is to come up with an interval estimate of population mean  based on the sample mean =280.

7. 7 Summary of Point Estimates of A Simple Random Sample of 121 Checking AccountsPopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean account balance account balance $310$306  = Population std. deviation for deviation for account balance account balance $66 s = Sample std. s = Sample std. deviation for deviation for account balance account balance$61 p = Population pro- portion of account portion of account balance no less than balance no less than $500 $500.3.27 = Sample mean = Sample mean account balance account balance = Sample pro- = Sample pro- portion of account portion of account balance no less than balance no less than $500 $500

8. 8 Interval Estimate of A Population Mean:  Known E ( ) =  Let’s first figure out the values of that provide the middle area about  of 95%.  The sample mean distribution of 121 checking account balances can be approximated by a normal distribution with E ( ) =  Let’s first figure out the values of that provide the middle area about  of 95%. 95% ab

9. 9 Interval Estimate of A Population Mean:  Known  Example: Checking Accounts  Given the middle area of 95%, we can find z values first and then convert z values to the corresponding values of. 95% a b -z 0.025 z 0.025 z 0 2.5 %

10. 10 Interval Estimate of A Population Mean:  Known  Example: Checking Accounts  Convert z values to the corresponding values of.

11. 11 Interval Estimate of A Population Mean:  Known  Example: Checking Accounts  We set the margin of error as. So, the interval estimate of population mean is. 95% a  b [--------- -----------] As long as falls between a and b, the interval will include the population mean. 

12. 12 Interval Estimate of A Population Mean:  Known  Example: Checking Accounts The rationale behind the interval estimate –  For any particular sample mean, we cannot compare it with the population mean  since  is unknown. But, what we are certain is that as long as falls between a and b. The interval will include the true value of .  In the example, = 306. So, the interval estimate of population account balance is. Because z 0.025 =1.96 and, the interval estimate is calculated as 306  1.96·6 = 306  11.76 or $294.24 to $317.76  We are 95% confident that will fall between a and b. So, the chance is 95% that the true value of  is no less than $294.24 and no more than $317.76.  On the other hand, there is a 5% chance that we make a mistake and the above interval estimate doesn’t include . Margin of Error

13. 13 Interval Estimate of A Population Mean:  Known  Interval Estimate of  where: is the sample mean 1 -  is the confidence level 1 -  is the confidence level 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

14. 14 Interval Estimate of A Population Mean:  Known  Values of z  /2 for the Most Commonly Used Confidence Levels 90%.10.05 1- .9500 1.645 90%.10.05 1- .9500 1.645 95%.05.025 1- .9750 1.960 95%.05.025 1- .9750 1.960 99%.01.005 1- .9950 2.576 99%.01.005 1- .9950 2.576 Confidence Area to the Level   /2 left of z  /2 z  /2 Level   /2 left of z  /2 z  /2

15. 15 Interval Estimate of A Population Mean:  Known  Example: Checking Accounts 90% 9.87 296.13 to 315.87 90% 9.87 296.13 to 315.87 Confidence Margin Level of Error Interval Estimate Level of Error Interval Estimate 95% 11.76 294.24 to 317.76 95% 11.76 294.24 to 317.76 99% 15.46 290.54 to 321.46 99% 15.46 290.54 to 321.46 The higher the confidence level, the wider the The higher the confidence level, the wider the Interval estimate.

16. 16 Interval Estimate of A Population Mean:  Unknown  is unknown, we will have to use the sample standard deviation s to estimate .  When  is unknown, we will have to use the sample standard deviation s to estimate .  is based on the t distribution. (See Table 2 of Appendix B in the textbook)  In this case, the interval estimate for  is based on the t distribution. (See Table 2 of Appendix B in the textbook) A specific t distribution depends on a parameter known as the degrees of freedom.A specific t distribution depends on a parameter known as the degrees of freedom. Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.Degrees of freedom refer to the number of independent pieces of information that go into the computation of s. As the degrees of freedom increases, t distribution is approaching closer to the Standard Normal Distribution.As the degrees of freedom increases, t distribution is approaching closer to the Standard Normal Distribution.

17. 17 t DistributionStandardnormaldistribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) of freedom) 0 z, t

18. 18 t Distribution  For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value.  The standard normal z values can be found in the infinite degrees (  ) row of the t distribution table. Standard normal z values 

19. 19 Interval Estimate of A Population Mean:  Unknown  Interval Estimate where: 1 -  = the confidence level 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 n = sample size n = sample size

20. 20 Interval Estimate of A Population Mean:  Unknown  Example: Consumer Age. The makers of a soft drink want to identify the average age of its consumers. A sample of 20 consumers was taken. The average age in the sample was 21 years with a standard deviation of 4 years. Construct a 95% confidence interval for the true average age of the consumers.

21. 21 Interval Estimate of A Population Mean:  Unknown  Example: Consumer Age At 95% confidence,  =.05, and  /2 =.025. At 95% confidence,  =.05, and  /2 =.025. In the t distribution table we see that t.025 = 2.093. t.025 is based on n - 1 = 20 - 1 = 19 degrees of freedom.

22. 22 Interval Estimate of A Population Mean:  Unknown  Example: Consumer Age Margin of Error We are 95% confident that the average age of the We are 95% confident that the average age of the soft drink consumers is between 19.13 and 22.87.

23. 23 Summary of Interval Estimation Procedures for a Population Mean Is the population standard deviation  deviation  known ? known ? Use YesNo Use  Known Case  Unknown Case Use the sample standard deviation s to estimate s

24. 24 Interval Estimate of A Population Proportion The general form of an interval estimate of a population proportion is

25. 25 Interval Estimate of A Population Proportion  Just as the sampling distribution of is key in estimating population mean, the sampling distribution of is crucial in estimating population proportion.  The sampling distribution of can be approximated by a normal distribution whenever np  5 and n(1-p)  5.

26. 26 Interval Estimate of A Population Proportion  /2  Normal Approximation of Sampling Distribution of Sampling distribution of of Sampling distribution of of pp 1 -   /2

27. 27 Interval Estimate of A Population Proportion  Interval Estimate of where: 1 -  is the confidence level 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

28. 28 Interval Estimate of A Population Mean:  Known  Example: Checking Accounts  Refer to our previous example of Checking Accounts. Out of the simple random sample of 121 accounts, the sample proportion of account balance no less than $500 is.27. Develop a 95% confidence interval estimate of the population proportion. where: n = 121, =.27, z  /2 = 1.96 We are 95% confident that the proportion of all checking accounts with a balance no less than $500 is between.23 and.31, which correctly includes the population proportion.30. Margin of Error