Lecture 41 Sections Wed, Nov 12, 2008

Transcription

1 Lecture 41 Sections Hampden-Sydney College Wed, Nov 12, 2008

2 Outline

3 one-proportion test that we just studied allows us to test a hypothesis concerning one proportion, or two categories, e.g., Obama and McCain. What if their are three categories, e.g., Obama, McCain, and Undecided? Or more than three categories? We need a way to compare the proportions in all the categories at the same time. That is, we need one statistic that summarizes all the differences.

4 one-proportion test that we just studied allows us to test a hypothesis concerning one proportion, or two categories, e.g., Obama and McCain. What if their are three categories, e.g., Obama, McCain, and Undecided? Or more than three categories? We need a way to compare the proportions in all the categories at the same time. That is, we need one statistic that summarizes all the differences.

5 one-proportion test that we just studied allows us to test a hypothesis concerning one proportion, or two categories, e.g., Obama and McCain. What if their are three categories, e.g., Obama, McCain, and Undecided? Or more than three categories? We need a way to compare the proportions in all the categories at the same time. That is, we need one statistic that summarizes all the differences.

6 one-proportion test that we just studied allows us to test a hypothesis concerning one proportion, or two categories, e.g., Obama and McCain. What if their are three categories, e.g., Obama, McCain, and Undecided? Or more than three categories? We need a way to compare the proportions in all the categories at the same time. That is, we need one statistic that summarizes all the differences.

7 one-proportion test that we just studied allows us to test a hypothesis concerning one proportion, or two categories, e.g., Obama and McCain. What if their are three categories, e.g., Obama, McCain, and Undecided? Or more than three categories? We need a way to compare the proportions in all the categories at the same time. That is, we need one statistic that summarizes all the differences.

8 Definition (Count data) Data that counts the number of observations that fall into each of several categories. data may be univariate or bivariate. Univariate example - Observe the color of a plain M&M candy. Bivariate example - Observe the color of an M&M candy and whether it is plain or peanut.

9 Univariate Example Observe the colors of plain M&M candies (2 pkgs). Color Count Blue 20 Orange 24 Green 25 Yellow 23 Brown 12 Red 10

10 Bivariate Example Observe the colors of M&M candies and whether they are plain or peanut. Plain Peanut Color (2 pkgs) (4 pkgs) Blue Orange Green Yellow Brown 12 7 Red 10 12

11 Two Basic Questions Basic Question for Univariate Data Do the data fit a specified distribution? For example, could these data have come from a uniform distribution? Basic Question for Bivariate Data Could the data in every category have come from the same distribution, whatever that distribution may be? For example, do plain and peanut M&Ms have the same distribution of colors?

12 Tests of Fit Definition (Goodness-of-fit Test) A hypothesis test that determines whether a set of data reasonably fits a specified distribution. goodness-of-fit test applies only to univariate data. null hypothesis specifies a discrete distribution for the population. We want to determine whether a sample from that population supports this hypothesis.

13 Observed and Expected Counts Definition (Observed Counts) observed counts are the counts that were actually observed in the sample. Definition (Expected Counts) expected counts are the counts that would be expected for a sample of that size if the null hypothesis were true.

14 Example If we selected 20 people from a group that was 60% male and 40% female, we would expect to get 12 males and 8 females. If we got 15 males and 5 females, would that indicate that our selection procedure was biased? What if we selected 100 people from the group and got 75 males and 25 females?

15 Example If we deal a 5-card poker hand, we should get One pair 42.26% of the time, Two pairs 9.51% of the time, Three of a kind 2.11% of the time, and Something else 46.12% of the time. Suppose a dealer deals 1000 poker hands and deals one pair 407 times, two pairs 122 times, three of a kind 36 times, and something else 435 times. Is this consistent with the theory? Or do we have reason to suspect him of some sort of chicanery?

16 Example Mars Candy Company reports that the colors of plain M&Ms are distributed as follows. Color Proportion Blue 24% Orange 20% Green 16% Yellow 14% Brown 13% Red 13%

17 Example I bought two packages yesterday and found Color Count Blue 20 Orange 24 Green 25 Yellow 23 Brown 12 Red 10

18 Example re were 114 candies in the two packages. Hypothetical Expected Observed Color Proportion Count Count Blue 24% Orange 20% Green 16% Yellow 14% Brown 13% Red 13%

19 Hypotheses null hypothesis specifies the probability (or proportion) for each color. null hypothesis is H 0 : p 1 = 0.24, p 2 = 0.20, p 3 = 0.16, p 4 = 0.14, p 5 = 0.13, p 6 = alternative hypothesis will always be a simple negation of H 0 : H 1 : H 0 is false. Let α = 0.05.

20 Expected Counts test statistic will involve the observed and the expected counts. To find the expected counts, we apply the hypothetical proportions to the sample size. For example, the hypothetical proportion for red is 24%, so we compute 24% of 114: = Do not round the values off to whole numbers.

21 Test Make a chart showing both the observed counts and the expected counts (in parentheses). Color Blue Orange Green Yellow Brown Red Observed (Expected) (27.36) (22.80) (18.24) (15.96) (14.82) (14.82)

22 Test Denote the observed counts by O and the expected counts by E. Define the chi-square (χ 2 ) statistic to be χ 2 = all cells (O E) 2. E

23 Value of the Test Clearly, if all of the deviations O E are small, then χ 2 will be small. But if even a few the deviations O E are large, then χ 2 will be large.

24 Value of the Test Now calculate χ 2. χ 2 = ( )2 ( )2 ( ) ( )2 ( )2 ( ) = =

25 Compute the p-value p-value is the likelihood of observing a χ 2 value as large at To find that value, we need to know something about the distribution of χ 2.

26 Degrees of Freedom χ 2 distribution has an associated degrees of freedom, just like the t distribution. Each χ 2 distribution has a slightly different shape, depending on the number of degrees of freedom. For example, we let χ 2 5 denote the chi-square statistic with 5 degrees of freedom. Definition (χ 2 degrees of freedom) In a goodness-of-fit test, the number of degrees of freedom is one less than the number of cells.

27 Degrees of Freedom Graph of χ

28 Degrees of Freedom Graph of χ

29 Degrees of Freedom Graph of χ

30 Degrees of Freedom Graph of χ

31 Degrees of Freedom Graph of χ

32 Degrees of Freedom Graph of χ

33 Degrees of Freedom Graph of χ

34 Degrees of Freedom Graph of χ

35 Properties of χ 2 chi-square distribution with df degrees of freedom has the following properties. χ 2 0. It is unimodal. It is skewed right (not symmetric!) µ χ 2 = df. σ χ 2 = 2df. If df is large, then χ 2 df is approximately normal with mean df and standard deviation 2df.

36 vs. Normal graph of χ 2 8 vs. N(8, 4)

37 vs. Normal graph of χ 2 32 vs. N(32, 8)

38 vs. Normal graph of χ vs. N(128, 16)

39 vs. Normal graph of χ vs. N(512, 32)

40 TI-83 Press 2nd DISTR. Select χ 2 cdf. Enter the lower endpoint, the upper endpoint, and the degrees of freedom. Press ENTER. probability appears in the display.

41 Practice Find P(χ 2 > 6) with df = 3. Find P(20 < χ 2 < 30) with df = 25. Find P(χ 2 < 10) with df = 6.

42 Homework Read Sections , pages Let s Do It! Exercises 1-5, page 928.