Exam 4 Review Exercises

Transcription

1 Math 160: Statistics Spring, 2014 Toews Exam 4 Review Exercises Instructions: Working in groups of 2-4, first review the goals and objectives for this exam (listed below) and then work the following problems. I ll field questions on this material today as you work, or on Wednesday during our last class session. Overview: This exam covers the following material: inference for a single proportion (8.1) inference for the difference between two proportions (8.2) goodness of fit (9.3) inference on two-way tables (9.1) basic data analysis on two-way tables (2.5) scatterplots (2.1) correlation (2.2) Objectives: The specific objectives of this test are to demonstrate your mastery of the following skills: using and understanding a variety of tables, including the standard normal, the binomial, and the chi square tables understanding how to estimate a population proportion p with a sample proportion ˆp understanding how the distribution of ˆp relates to the distribution of a binomial random variable, and understanding how a normal distribution can be used to approximate this binomial distribution forming confidence intervals for proportions (single sample) and differences of proportions (two sample) calculating P values for proportions and differences of proportions understanding how to conduct and interpret a goodness of fit test (i.e. inference on a two-way table) understanding how to interpret and calculate joint, marginal, and conditional distributions from count data in a two-way table understanding how form and interpret scatterplots of two quantitative variables measured on a single population understanding how to calculate and interpret the correlation coefficient

2 1. Mark the following as T for true or F for false. (Don t guess: make sure you understand the underlying issues.) T F Two way tables are used to represent categorical data, not quantitative data T F You can calculate confidence intervals and P values for two-way tables. T F The alternative hypothesis in goodness of fit tests can be one sided or two sided. T F When doing inference on two-way tables, the null hypothesis is always of the form no association between variables. T F If two variables X and Y have correlaiton r = 0, this means that there is no relation between X and Y. T F If two variables X and Y have correlaiton r = 1, this means that the scatterplot of X against Y looks like a line. T F Suppose some proportion p of a population satisfies some property of interest. Define X to be the random variable given by drawing an SRS of size n from this population and then setting X equal to the count of individuals (in the sample) who satisfy this property. Then X is exactly binomial, with parameters n and p. T F Consider the random variable X defined in the previous problem. If ˆp = X/n, then ˆp is distributed approximately N(p, p(1 p)/n), as long as np 10 and n(1 p) Estimating proportions (a) Suppose you are interested in estimating the percentage of UPS students who support looser gun laws. You take an SRS of size 20 and find that exactly 1 person supports looser gun laws. Estimate p, the proportion of UPS students who support looser gun laws. (b) Explain why you can t use the formula ˆp ± z ˆp(1 ˆp) n to provide a confidence interval for this estimate. (Hint: this formula is based on the normal approximation for binomials. Can you use the normal approximation in this problem?)

3 (c) Now suppose you sample 200 students and find that 10 of them support looser gun laws. Provide a 98% confidence interval for p. (d) Now suppose you sample 300 students from Seattle U. and find that 20 of them support looser gun laws. Estimate the difference between the rates of support for looser guns laws at these two universities, and provide a 98% confidence bound for your estimate. (e) Suppose your friend claims that the rate of support for looser gun laws is the same at every university. You disagree: you think that a non-sectarian university like UPS might have even less support than a religiously affiliated school like Seattle U. Formulate appropriate null and alternative hypotheses, and calculate a P value (based on the data above.) Can you reject the null hypothesis at the 5% level?

4 3. Goodness of Fit You have decided to collect data on the kinds of birds that come to your feeder every Spring. Your gut instinct is that 35% of the birds are bluejays, 25% are robins, 20% are woodpeckers, and the rest are unknown. You collect the the following data: Type Count Bluejays 120 Robins 90 Woodpeckers 60 Unknown 80 You decide to use this data to do a goodness of fit test to see if this data is compatible with the percentages that you assumed. (a) Write down an appropriate null hypothesis. (b) Calculate X 2, the statistic you need to perform the hypothesis test. How is this statistic distributed? (c) Calculate the P value of your data. Can you reject the null hypthesis at the 1% level?

5 4. Inference on two-way tables A biologist has been taking data on pit monkies to determine whether or not there is an association between the presence of a certain gene and the color of the tuft on the tail. For each pit monkey from a sample, the biologist records the presence or absence of the gene, as well as the color of the tail tuft. The data is as follows: Observed Counts Has gene Lacks gene Red tuft Brown tuft White tuft 10 5 No tuft 5 20 Having taken the data, the biologist would now like to do inference to see if there really is an association between the presence of the gene and the color of the tuft. (a) Write down an appropriate null hypotheis. (b) What is the implied alternative hypothesis? (c) Calculate the expected table of counts under this null hypothsis. Expected counts Has gene Lacks gene Red tuft Brown tuft White tuft No tuft (d) Calculate X 2, the statistic you need to do inference on these count data. How is X 2 distributed? (e) Calculate the P value for your data. Can you reject your H 0 at the 5% level?

6 5. Distributions derived from tables For this problem, we use the table from problem 4: Observed Counts Has gene Lacks gene row sum Red tuft Brown tuft White tuft No tuft column sum (a) Let X be the random variable of tuft color and Y the random variable of having the gene. Compute the joint distribution of X and Y. (b) Compute the marginal distributions of X and Y. (c) Compute the conditional distribution of X given Y, and the conditional distribution of Y given X.

7 6. Scatterplots and Correlation A statistics professor at a big university gives two exams each semester. She wonders if there is any relation between scores on the first exam and scores on the second. To answer this question, she gathers data on five students. Her data is as follows: Test Test (a) Plot the data in a scatterplot. (b) Comment on the form, direction, and strength of the association between the two variables. (c) Calculate the correlation of the data. Based on the shape of the scatterplot, does your answer surprise you?