United Stats Of AMERICA

Unit 7 chapters Jordo, Rob III, Kins and Toph

Chapter 26 Three types of Chi-squared tests: 1) Goodness of Fit 2) Homogeneity 3) Independence

Goodness gracious of Fit The test is used when you have one categorical variable from a single population. It is used to determine whether sample data are consistent with a hypothesized distribution. (How “good” the data fit the hypothesis)

Goodness of Fit Conditions: The sampling method is random. The variable under study is categorical. (counted) The expected value of the number of sample observations in each level of the variable is at least 5. (expected cells > 5) Degrees of freedom: df = n - 1 n = total number of categories

Goodness of Fit Hypothesis: We need a null hypothesis (H 0 ) and an alternative hypothesis (H a ). The hypotheses are mutually exclusive. So if one is true, the other must be false; and vice versa.null hypothesis alternative hypothesis For a chi-square goodness of fit test, the hypotheses take the following form. H 0 : The data are consistent with a specified distribution. H a : The data are not consistent with a specified distribution.

Goodness of Fit Acme Toy Company prints baseball cards. The company claims that 30% of the cards are rookies, 60% veterans, and 10% are All-Stars. The cards are sold in packages of 100. Suppose a randomly-selected package of cards has 50 rookies, 45 veterans, and 5 All-Stars. Is this consistent with Acme's claim? Use a 0.05 level of significance. Here you can see there is only one categorical variable, and putting these numbers in the calculator and doing a x^2 GOF test is super easy. RookiesVeteransAll-Stars

no Homogeneity This test is used for one categorical variable from two populations. It is used to determine whether frequency is consistent across different populations.

Homogeneity Conditions ●Expected Cells > 5 ●Categorical ●Random

Homogeneity Hypothesis: H 0 : The distribution of separate categories is the same. H a : The distribution is different.

Homogeneity Viewing PreferencesRow total Lone Ranger Sesame Street The Simpsons Boys Girls Column total In a study of the television viewing habits of children, a developmental psychologist selects a random sample of 300 first graders boys and 200 girls. Each child is asked which of the following TV programs they like best: The Lone Ranger, Sesame Street, or The Simpsons. Results are shown in the contingency table above. Do the boys' preferences for these TV programs differ significantly from the girls' preferences? Use a 0.05 level of significance. contingency table

declaration of Independence ●We use Independence to find out if one thing causes another or if two samples relate. ●Ho: will always be that X is INDEPENDENT of Y. ●Ha: will always be X is DEPENDENT OF Y.

Independence Example: YesNoTotal Male268 Female4812 Total61420 ●To find Degrees of freedom you would do: (number of rows-1)(number of columns-1) For this chart it would be (2-1)(2-1)=1 ●We have to find expected cells to make sure that they are greater or equal to 5. To do this for the shaded cell we would do 6 times 8 divided by 20 which equals 2.4 which is less than five so this example would not work.

Independence Conditions: -Categorical -Counted -Expected > or to 5 -Random -Independent

Chapter 27 Regression Analysis

●We use regression analysis to determine if a relationship exists between two quantitative variables. ●Chapter 27 is a throwback to earlier chapters ○Chapter 8 - Scatterplots H 0 : 1=0 (This means that the slope is equal to 0, meaning that there is no linear relationship) H A : 1 ≉ 0 (The slope is not equal to 0, so there is a linear relationship)

Conditions ●Straight Enough (Linear) ●Quantitative Data ●Residual Graph is good ●Random ●Nearly Normal ●No Outliers

Example How to make regression equation: ●The row labeled “Constant” or the name of the y-variable is the information for the y- intercept. (Beta 0) ●The other row, which is usually labeled with the name of the x-variable, shows the slope. (Beta 1) Ŷ= (x) *****Make sure you talk about the slope and the r-squared in context*****

Making an inference Since we are testing the Beta 1, which is slope, we will look at P for the x variable. ●P=0.000 ●Conclusion: We have enough evidence to reject the null, and can conclude that there is a relationship between the two variables.

Confidence Intervals The equation for a confidence interval is: 1 士 T*(SE) 1 is SE is We’ll do a 95% confidence interval, so we’ll need to find the t-score using an inverse-T function on the calculator. The equation comes out to be 士 2.1(.3842). Conclusion: We can be 95% confident that the true mean of the relationship between the two variables is between and *****Degrees of Freedom are always n-2*****