1. Chapter 8Interval Estimation

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

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. 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( = m) = 0. • 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. Margin of Error and the Interval Estimate • The general form of an interval estimates of a population mean is

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. = Sample mean account balance = Sample pro- portion of account balance no less than $500 Summary of Point Estimates of A Simple Random Sample of 121 Checking Accounts Population Parameter Parameter Value Point Estimator Point Estimate m = Population mean account balance $310 $306 $66 s = Sample std. deviation for account balance $61 s = Population std. deviation for account balance .3 .27 p = Population pro- portion of account balance no less than $500

8. Interval Estimate of A Population Mean: Known • 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% a b

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% 95% 2.5% 2.5% z -z0.025 z0.025 a 0 b

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

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 . As long as falls between a and b, the interval will include the population mean .  95%  a b [--------- -----------] [--------- -----------]

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 z0.025=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. Interval Estimate of A Population Mean: Known • Interval Estimate of m where: is the sample mean 1 - is the confidence level z/2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution s is the population standard deviation n is the sample size

14. Interval Estimate of A Population Mean: Known • Values of za/2 for the Most Commonly Used Confidence Levels Confidence Area to the Level aa/2 left of za/2 za/2 90% .10 .05 1- a/2= .9500 1.645 95% .05 .025 1- a/2= .9750 1.960 99% .01 .005 1- a/2= .9950 2.576

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

16. Interval Estimate of A Population Mean: Unknown • When s is unknown, we will have to use the sample standard deviation s to estimate s . • In this case, the interval estimate for m 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. • 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.

17. t Distribution t distribution (20 degrees of freedom) Standard normal distribution t distribution (10 degrees of freedom) z, t 0

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. Interval Estimate of A Population Mean: Unknown • Interval Estimate where: 1 - = the confidence level t/2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation n = sample size

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

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

23. Summary of Interval Estimation Proceduresfor a Population Mean Is the population standard deviation s known ? Yes No Use the sample standard deviation s to estimate s s Known Case Use Use s Unknown Case

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

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. Interval Estimate of A Population Proportion • Normal Approximation of Sampling Distribution of Sampling distribution of /2 /2 1 -  p

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 /2 in the upper tail of the standard normal probability distribution is the sample proportion

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 Margin of Error 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.