1 STA100C fall 2016 Homework 9 Due at the beginning of class on Monday November 28, 2016 Note new due date and change of due dates for remaining homeworks homework due 9 Monday 11/28/16 10 Friday 12/02/16 Points lost if you don't follow the rule all points (no credit) Write your section number on the right side at the top of the first page. MAKE SURE YOU WRITE THE SECTION YOU ARE REGISTERED FOR Write your name and student ID 1 on the left side at the top of the first page. Your name should be spelled exactly the same as it is on SmartSite all points (no credit) (no nicknames please) Write the homework number in the middle at the top 1 of the first page. (Any format is fine, for example, HW1, homework 1, etc.) Staple all pages. 1 If your homework is on paper pulled out of a notebook, cut off all of the fringes (from the torn horizontal threads that attached the paper to the notebook). 1 For example, for homework 9 if your name is John Smith, your student ID is 123456789, and you are in section C01, then the top of your first page should look like this John Smith 123456789 Homework 9 C01 Be kind to the grader. make sure you write your name clearly (so it is easy to read) write neatly circle all final answers (so they are easy to find) Round all test statistics to two digits after the decimal point. Round all p-values to four digits after the decimal point. You should do the rounding after the last step in the calculations. If you round in intermediate steps the rounding error tends to accumulate. You should do most of the calculations in this assignment without using R (you will need R to get the p-values). However, if you want to check your chi-square tests with R there is some R code for chi-square tests on SmartSite under Resources R resources chi square test in R.pdf 2 1. Scientists have used Mongolian gerbils when conducting neurological research. A certain breed of these gerbils was crossed and gave progeny of the following colors. color number of progeny black 40 brown 59 white 42 Are these data consistent with the 1:2:1 ratio predicted by a certain genetic model? Use a chi-square test with = 0.05. (a) Calculate the expected numbers of black, brown, and white gerbils assuming that the null hypothesis that the ratio is 1:2:1 is true. (b) Calculate the chi-square test statistic. (c) If = 0.05, what is the critical value? You can either get this from table 9 in your book (also on SmartSite) or use the following R command. qchisq(1-alpha,degf) where alpha is the alpha level and degf is the degrees of freedom. (d) Use the chi-square table (table 9 in your book and also on SmartSite) to find an interval contains the p-value. (See page 319 of lecture notes.) (e) Calculate a p-value using the following R command. (1-pchisq(TS,df=degf)) where TS is the test statistic and degf is the degrees of freedom. Note that we don't multiply by 2 because the chi-square test is always testing the null against a non-directional alternative. (f) At = 0.05, do the data provide evidence the genetic model does not hold? How did you make your decision? 2. Geneticists studying the inheritance pattern of cowpea plants classified the plants in one experiment according to the nature of their leaves. The data are as follows: type number I 179 II III 44 23 Test the null hypothesis that the three types occur with probabilities 12/16, 3/16, and 1/16. Use a chi-squared test with = 0.1 (a) Calculate the expected numbers of types I, II, and III assuming the null hypothesis is true. (b) Calculate the chi-square test statistic. (c) If = 0.1, what is the critical value? (d) Use the chi-square table (table 9 in your book and also on SmartSite) to find an interval contains the p-value. (See page 319 of lecture notes.) (e) Calculate the p-value. (f) At = 0.1 do the data provide evidence against the null hypothesis? How did you make your decision? 3 3. A survey is conducted in a large high to study the relationship between sex and alcohol. Results are given in the following table. drinkers males 58 females 46 total 104 non-drinkers 42 54 96 total 100 100 200 Define p1 as the proportion of male and p2 as the proportion of female students in this high school who consume alcohol on a regular basis. Use a chi-square test to test the null hypothesis H0 : p1 = p2 against the alternative hypothesis HA : p1 6= p2 . (a) Find the expected number of males who are drinkers assuming the null hypothesis of no difference between males and females is true. Note that given this one expected number the three other expected numbers can be found by subtraction from either the row or column totals. (b) Calculate the chi-square test statistic. (c) What are the degrees of freedom for this test? (d) If = 0.1, what is the critical value? (e) Use the chi-square table (table 9 in your book and also on SmartSite) to find an interval contains the p-value. (See page 319 of lecture notes.) (f) Calculate the p-value. (g) At = 0.1 do the data provide evidence against the null hypothesis? How did you make your decision? 4. For a study of a population of the fruitfly Drosophila subobscura, researchers placed baited traps in two woodland sites and one open-ground area. The numbers of male and female flies trapped in a single day are given below. woodland site 1 males 89 females 31 total 120 woodland site 2 34 20 54 open ground total 74 197 136 187 210 384 Use a chi-square test with = 0.05 to compare the sex ratios at the three sites. The null and alternative hypotheses are H0 : sex and site are independent HA : sex and site are not independent (a) Calculate the chi-square test statistic. (b) What are the degrees of freedom for the test? (c) If = 0.05, do the data provide evidence in favor of the alternative hypothesis. How did you make your decision