Confidence Intervals

1. Confidence Intervals Week 10 Chapter 6.1, 6.2

2. What is this unit all about? • Have you ever estimated something and tossed in a “give or take a few” after it? • Maybe you told a person a range in which you believe a certain value fell into. • Have you ever see a survey or poll done, and at the end it says: +/- 5 points. • These are all examples of where we are going in this section.

3. Chapter 6.1 - disclaimer • To make this unit as painless as possible, I will show the formula but will teach this unit with the use of the TI – 83 graphing calculator whenever possible. • It is not always possible to use the TI-83 for every problem. • You can also follow along in Chp. 6.1 in the TEXT and use their examples in the book to learn how to do them by hand.

4. What is a Confidence Interval? • If I were to do a study or a survey, but could not survey the entire population, I would do it by sampling. • The larger the sample, the closer the results will be to the actual population. • A confidence interval is a point of estimate (mean of my sample) “plus or minus” the margin of error.

5. What will we need to do these? • Point of estimate – mean of the random sample used to do the study. • Confidence Level – percentage of accuracy we need to have to do our study. • Critical two-tailed Z value - (z-score) using table IV. • Margin of Error – a formula used involving the Z value and the sample size.

6. Formula for Confidence Intervals * This formula is to be used when the Mean and Standard Deviation are known:

7. Finding a Critical Z-value (Ex 1) – Find the critical two-tailed z value for a 90% confidence level: * This means there is 5% on each tail of the curve, the area under the curve in the middle is 90%. Do Z (1-.05) = Z .9500 *We will be finding the z score to the left of .9500 in table IV. It lands in-between .9495-.9505, thus it is = +/- 1.645 (this is the 5% on each end)

8. Finding a Critical Z-value (Ex 2) – Find the critical two-tailed z value for a 95% confidence level: * This means there is 2.5% on each tail of the curve, the area under the curve in the middle is 95%. Do Z (1-.025) = Z .9750 *We will be finding the z score to the left of .9750 in table IV. * It is = +/- 1.96 (this is the 2.5% on each end)

9. Finding a Critical Z-value (Ex 3) – Find the critical two-tailed z value for a 99% confidence level: * This means there is .005% on each tail of the curve, the area under the curve in the middle is 99%. Do Z (1-.005) = Z .9950 *We will be finding the z score to the left of .9950 in table IV. * It is = +/- 2.575 (this is the .005% on each end)

10. Finding a Critical Z-value (Ex 4) – Find the critical two-tailed z value for a 85% confidence level:

11. Margin of Error • The confidence interval is the sample mean, plus or minus the margin of error.

12. Find the MoE: Ex (5) – After performing a survey from a sample of 50 mall customers, the results had a standard deviation of 12. Find the MoE for a 95% confidence level.

13. Special features of Confidence Intervals • As the level of confidence (%) goes up, the margin of error also goes up! • As you increase the sample size, the margin of error goes down. • To reduce the margin of error, reduce the confidence level and/or increase the sample size. • If you were able to include the ENTIRE population, the would not be a margin of error. • The magic number is 30 samples to be considered an adequate sample size.

14. Finding Confidence Intervals: (Ex 6) – After sampling 30 Statistics students at NCCC, Bob found a point estimate of an 81% on Test # 3, with a standard deviation of 8.2. He wishes to construct a 90% confidence interval for this data.

15. How did we get that?

16. Using TI-83 to do this: • Click STAT • go over to TESTS • Click ZInterval • Using the stats feature, input S.D., Mean, sample size, and confidence level. • arrow down, and click enter on calculate.

17. Finding Confidence Intervals: (Ex 7) – After sampling 100 cars on the I-90, Joe found a point estimate speed 61 mph and a standard deviation of 7.2 mph. He wishes to construct a 99% confidence interval for this data.

18. Finding an appropriate sample size • This will be used to achieve a specific confidence level for your study.

19. Find a sample size: (Ex 8) – Bob wants to get a more accurate idea of the average on Stats Test # 3 of all NCCC stats class students . How large of a sample will he need to be within 2 percentage points (margin of error), at a 95% confidence level, assuming we know the σ = 9.4?

20. How did we get this?

21. Finish Bob’s Study: Ex (9) - Now lets say Bob wants to perform his study, finds the point of estimate for Test # 3 = 83, with a SD of 9.4 and confidence level of 95%. Find the confidence interval for this study.

22. What about an interval found with a small sample size? (chp 6.2) • To do these problems we will need: • TABLE 5: t-Distribution. • Determine from the problem: n, x, s. Sample, mean, sample standard deviation. • Use the MoE formula for small samples: t-value from Table 5 d.f. = n-1 (degrees of freedom)

23. Small Sample Confidence Int. (Ex 10) – Trying to determine the class average for Test # 3, Janet asks 5 students their grade on the test. She found a mean of 78% with a σ = 7.6. Construct a confidence interval for her data at a 90% confidence level.

24. What did we do? d.f. = 5-1 = 4; .90 Lc = 2.132

25. Or with TI-83/84 STAT TESTS 8:TInterval Stats Input each value, hit calculate.