1. 1 Where we are going : a graphic. 1 2 Paired 2 or more Means Proportions Variances Categories Slopes Ho: / CI Samples Ho: / CI Ho: Ho: / CI

2. 2 Where we are going : some words. Statistics is an organized way to evaluate information (numerical and otherwise). We will work through the mechanics of hypothesis testing and confidence interval estimation (except for categorical) in the 11 situations on the previous page. We will do this by hand and with the computer. The goal is to review the procedures; to discuss when they are (and when they are not) appropriate (and a bit about why); and to provide you with computer skills necessary to accomplish the arithmetic.

3. 3 Begin: 15 second resting pulse rate readings were observed on 128 students. First: usual measures of location, dispersion, and the stem and leaf graph. Next: SAS code and output are given in Example 1. Comment: These slides are not meant to be detailed, but a brief outline of material. Details are given in the text, notes, and lecture.

4. 4 2021132120172021 2020222118211823 2122191722181818 2021152620192221 1921202319201618 1621161622212121 2120222220212022 1818222021191918 1623162421192018 2019162121221517 2217182219201918 2220202118152122 2220181422201816 2018212220191822 2219191918191920 1920232116142219 1. List the data:measures of Location descriptive but inadequate for obvious reasons

5. 5 2. Sort the data, then list it: better, still too much information. Median (20 = avg of 20 & 20 ) is easily determined. 1317181920212122 1417181920212122 1518181920212122 1518191920212122 1618191920212222 1618192020212222 1618192020212223 1618192020212224 17181920 21212226

6. 6 2 6 2 2 4 2 3333 2 22222222222222222222 2 11111111111111111111111 2 000000000000000000000000 1 99999999999999999 1 88888888888888888 1 7777 1 666666666 1 555 1 44 1 3 3. Stem and leaf: first digit is the stem, second is the leaf. Useful: Mode (20, most common) easily determined

7. 7 4. The sample mean: Location: used a lot.

8. 8 freq 2 6 1 2 0 2 4 1 2 3333 4 2 22222222222222222222 20 2 11111111111111111111111 23 2 000000000000000000000000 24 1 99999999999999999 18 1 88888888888888888 18 1 7777 4 1 666666666 9 1 555 3 1 44 2 1 3 1 5. Stem and leaf.2 : Median, Mean, and Mode identified. *---------* median 20 + mean 19.60

9. 9 6. Range: a measure of spread or dispersion Range = (Maximum - Minimum) = 26-13 = 13 7. Inner Quartile Range (IQR): A. The median divides the sample into halves. B. Compute the median of each half: Lower half: (average of 18 & 18 = 18) Upper half: (average of 21 & 21 = 21) C. These two numbers define (approximately) the two inner quarters. The IQR is the distance between them. IQR = 21-18 = 3

10. 10 freq 2 6 1 2 0 2 4 1 2 3333 4 2 22222222222222222222 20 2 11111111111111111111111 23 2 000000000000000000000000 24 1 99999999999999999 18 1 88888888888888888 18 1 7777 4 1 666666666 9 1 555 3 1 44 2 1 3 1 8. Stem and leaf.3 : Median, Mean, Mode, and IQR identified. +----------+ Q3 = 21 | *---------* median 20 | + | | | mean 19.60 +----------+ Q1 = 18

11. 11 9. Fences and outliers: A. the IQR defines the middle 50% of the sample. B. Inner fences are constructed as: Q1 - 1.5(IQR) = 18 - 1.5(3) = 13.5 Q3 + 1.5(IQR) = 21 + 1.5(3) = 25.5 These are the ‘whiskers’ on the box plot. Observations outside this fence are ‘mild’ (O). C. Outer fences are constructed as: Q1 - 3(IQR) = 18 - 3(3) = 9 Q3 + 3(IQR) = 21 + 3(3) = 30 Observations outside this fence are ‘extreme’ (*).

12. 12 freq 2 6 1 2 0 2 4 1 2 3333 4 2 22222222222222222222 20 2 11111111111111111111111 23 2 000000000000000000000000 24 1 99999999999999999 18 1 88888888888888888 18 1 7777 4 1 666666666 9 1 555 3 1 44 2 1 3 1 10. Stem and leaf.4 : Median, Mean, Mode, IQR, inner fences,and mild outliers identified. 0 _ | +----------+ Q3 = 21 | *----------* median 20 | + | | | mean 19.60 +----------+ | Q1 = 18 | - 0

13. 13 11. The sample variance: dispersion: used a lot. Computations: (memorize these)

14. 14 12. The sample variance: example

15. 15 13. The sample variance: a graphic 1 2 4 5 5 4 2 1 y y Variance is the average area of the ‘squares’

16. 16 14. The sample variance: pulse rate data: Using the formula:

17. 17 15. The sample variance: pulse rate data again Using the formula:

18. 18 16. The sample standard deviation: Standard deviation is the square root of variance For the pulse rate data: S = = 2.2390

19. 19 17. Practice these calculations !: Write down a few numbers or take a few measurements and compute: A. Median, Mode, and Mean B. a stem and leaf graph C. Range, Variance, and Standard Deviation. For example: 2 9 5 3 7 3 5 3 or the digits in your SSOC number, or telephone number. This is the end of set 1.