Confidence Intervals Week 10 Chapter 6.1, 6.2

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.

Chapter 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.

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.

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.

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

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 , thus it is = +/ (this is the 5% on each end)

(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 = +/ (this is the 2.5% on each end) 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 = +/ (this is the.005% on each end) Finding a Critical Z-value

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

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

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.

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.

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.

How did we get that?

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.

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.

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

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?

How did we get this?

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.

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)

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.

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

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