Statitics And probability:-
\fBeliefs of college freshmen Every year, a large-scale poll of college freshmen conducted by the Higher Education Research Institute at UCLA asks their opinions about a variety of issues. In 2004, although women were more likely to rate their time management skills as "above average, they were also twice as likely as men to indicate that they frequently feel overwhelmed by all they have to do (36.4% versus 16.3%). a. If results for the population of college freshmen were similar to these, would gender and feelings of being overwhelmed be independent or dependent? b. Give an example of hypothetical population percentages for which these variables would be independent.Sample evidence about independence Refer to the exercise 1. Go to the GSS Web site and construct a table relating happiness (HAPPY) to the variable you chose (AFTERLIF, FINRELA. HEALTH, REGION, or JOBSAT). Inspect the conditional distributions and indicate whether independence seems plausible, with the sample conditional distributions all being quite similar. Exercise 1 What is independent of happiness? Which one of the following variables would you think most likely to be independent of happiness: belief in an afterlife, family income, quality of health, region of the country in which you live, satisfaction with job? Explain the basis of your reasoning.Life after death and gender In the 2012 GSS, 605 of 790 males and 822 of 977 females indicated a belief in life after death. (Source: Data from CSM. UC Berkeley.) a. Construct a 2 x 2 contingency table relating gender of respondent (SEX, categories male and female) as the rows to belief about life after death (POSTLIFE, categories yes and no) as the columns. b. Find the four expected cell counts when assuming independence. Compare them to the observed cell counts, identifying cells having more observations than expected. c. For this data, X" = 18.0. Verify this value by plugging into the formula for X" and computing the Sum.Smoking and alcohol Refer to the exercise 1. A similar table relates cigarette use to alcohol use. The MINITAB output for the chi-squared test follows. a. True or false: If we use cigarette use as the column variable instead of alcohol use, then we will get different values for I he chi-squared statistic and the P-value shown in the table. b. Explain what value you would get for the z statistic and P-value if you conducted a significance test of HQ: p1 = p2 against Ha: p1 # p2, where p1 is the population proportion of non cigarette users who have drunk alcohol and p2 is the population proportion of cigarette users who have drunk alcohol Dayton student survey Row: cigarette Columns: alcohol no yes no 281 500 yes 45 1449 Pearson Chi-Square = 451.404, DF =1, P-Value = 0.000 Exercise 1 Cigarettes and marijuana The table on the following page refers to a survey1 in which senior high school students in Dayton, Ohio. were randomly sampled. It cross-tabulates whether a student had ever smoked cigarettes and whether a student had ever used marijuana. Analyze these data by (a) finding and interpreting conditional distributions with marijuana use as the response variable and (b) reporting all five steps of the chi-squared test of independence. Marijuana Cigarettes Yes No Yes 914 581 No 45 735 1 Source: Data from personal communication from Harry Khamis. Wright State University.\fTesting a genetic theory In an experiment on chlorophyll inheritance in corn, for 1103 seedlings of self-fertilized heterozygous green plants, 854 seedlings were green and 249 were yellow. Theory predicts that 75%% of the seedlings would be green. a. Specify a null hypothesis for testing the theory. b. Find the value of the chi-squared goodness-of-fit statistic and report its of c. Report the P-value, and interpret.Birthdays by quarters Based on a random sample of 84 students taken at Williams College, the table shows how many had their birthday in the first, second, third, or fourth quarter of the year. Is there evidence that the probabilities of having a birthday in a given quarter are not equal? Birthday Jan-Mar Apr-Jun Jul-Sep Oct-Dec 19 21 31 13 84 a. Formulate the null and alternative hypotheses. b. Find the expected values and compute the chi-squared statistic, either by hand or using software. c. The test will be based on how many degrees of freedom? d. Find the. P-value and write a conclusion in context.Checking a roulette wheel Karl Pearson devised the chisquared goodness-of-fit test partly to analyze data from an experiment to analyze whether a particular roulette wheel in Monte Carlo was fair, in the sense that each outcome was equally likely in a spin of the wheel. For a given European roulette wheel with 37 pockets (with numbers 0. 1, 2. ... . 36), consider the null hypothesis that the wheel is fair. a. For the null hypothesis, what is the probability for each pocket? b. For an experiment with 3700 spins of the roulette wheel, find the expected number of times each pocket is selected. C. In the experiment, the 0 pocket occurred 110 times. Show the contribution to the X2 statistic of the results for this pocket. d. Comparing the observed and expected counts for all 37 pockets, we get X2 = 34.4 Specify the of value, and indicate whether there is strong evidence that the roulette wheel is not balanced. (Hint: Recall that the dfvalue is the mean of the distribution.)Concerned about global warming? The Institute for Public Opinion Research at Florida International University has conducted the FIUFlorida Poll (www2.fiu.edw/orgs/ipor/globwarm2.htm) of about 1200 Floridians annually since 1988 to track opinions on a wide variety of issues. In 2006 the poll asked, "How concerned are you about the problem of global warming?" The possible responses were very concerned, somewhat concerned, not very concerned, and haven't heard about it. The poll reported percentages (44, 30, 21. 6) in these categories. a. Identify the sample and the population. b. Are the percentages quoted statistics or parameters? Why
