Practice First Midterm Exam

Name: Math 17 – Introduction to Statistics PRACTICE First Midterm Exam Instructions: 1. Show all work. You may receive partial credit for partially completed problems. 2. You may use calculators and a one-sided sheet of reference notes, as well as the provided table. You may not use any other references or any texts. 3. You may not discuss the exam with anyone but me. 4. Suggestion: Read all questions before beginning and complete the ones you know best first. Point values per problem are displayed below if that helps you allocate your time among problems. 5. Good luck! Problem 1 2 3 4 Total 12 11 12 15 50 Points Earned Possible Points 1. Fifty-five parents of grade school children were interviewed regarding the breakfast habits of their children. Two questions of interest were “Do your children eat breakfast?” and “What was your child’s average grade on the state standardized test?” Responses to both questions were recorded as yes or no for breakfast and values between 0-10 for grade. Grades for the breakfast and no breakfast groups were summarized using a boxplot. a. Is this an experiment or an observational study? b. The response variable in this situation is _______________ while the explanatory variable is ________________. c. Breakfast is an example of a (choose one) categorical quantitative variable. d. Based on the boxplots, which breakfast group performed better on the standardized test? How can you tell? e. What is (approximately) the IQR for the grade scores of children who do eat breakfast? f. No outliers are visible on the boxplot. Explain why there are no outliers plotted specifically for the group of children who do not eat breakfast. g. The lowest score for the group of children who do eat breakfast is roughly equal to the ________________ for the group of children who do not eat breakfast. h. If some of the selected parents for the original sample declined to answer both questions, what sort of bias would have occurred? selection bias response bias nonresponse bias 2. Julia and Dennis have collected a data set from elementary Grade\Pref Vanilla school children at a summer camp on their ice cream First 12 preference. They recorded each student’s grade Third 48 (first,third,fifth, no second or fourth grade students) and Fifth 24 their ice cream preference (choosing between vanilla and Total chocolate). The following table summarizes the results. You may treat the probabilities resulting from the table as population probabilities. Chocolate 4 16 8 Total a. Compute the following probabilities: i. The probability a randomly selected summer camp child is in the third grade ii. The probability a randomly selected summer camp child preferred chocolate ice cream iii. The probability that a randomly selected summer camp child is in the fifth grade given that he/she prefers chocolate ice cream iv. The probability that a randomly selected summer camp child is in the third grade given that he/she prefers vanilla ice cream b. Are third grade status and ice cream preference independent? How can you tell? c. If you choose 2 summer camp children at random (assume random sample), what is the probability that at least one is in the third grade? d. Assume the summer camp children have to wait in line for components for Smores at the campfire. Assume the distribution of waiting times is uniform from 3 minutes to 15 minutes. Let X be waiting time. i. What is the expected value for waiting time in line? ii. What is the probability a child waits between 3 and 6 minutes for their Smore components? iii. Name another continuous distribution besides a uniform distribution. 3. A researcher studying the heights of certain shrubs in the forest has collected a sample of 50 shrub heights in feet (continuous numerical variable). However, he is unsure of how to proceed with any analysis, but he makes a histogram and a Q-Q plot of the data. a. What would you tell the researcher to do as a preliminary analysis? (Be specific – what graphs, etc.) b. The sample standard deviation of shrub heights was 1.34 with a sample mean of 4.32. Interpret the standard deviation. c. Based on the two plots the researcher made, what can you conclude about the shape of the distribution of the shrub heights? Why? d. The researcher wants to examine the sample mean shrub height. What is the distribution of the sample mean shrub height for a sample of size 50 if the population mean shrub height is 4 and the population variance is 2? What result allows you to provide this distribution? e. It turns out that the researcher’s original sample of heights was based on shrubs in a 1 km square block within a biological research zone. The research zone has 16 such square kilometer blocks with 4 blocks at each of four altitudes. The researcher plans to obtain a larger sample of shrub heights, say 256 heights. From previous research, you know there is an effect of altitude on shrub height. Describe a sampling plan that you might use if you were the researcher and you wanted to deal with this effect when sampling. 4. Vitamin D deficiency has recently re-entered the public health scene as a serious problem among American adults. The current recommendation is to have 400 IU/day of Vitamin D. Assume that in a local community, the amount of Vitamin D ingested has a normal distribution with population mean 400 and standard deviation 40. a. Ted and Janice are a happily married couple living in this community. Ted ingests 350 units of Vitamin D daily while Janice ingests 420 units a day. Who has the more unusual Vitamin D intake? How can you tell? b. 20% of members of this community ingest under what amount of Vitamin D daily? c. Assume that nationwide, 80% of adult Americans are Vitamin D deficient. A random sample of 60 adult Americans is selected. i. What is the sampling distribution of the sample proportion of adult Americans who are Vitamin D deficient for a sample of size 60? Include the check that this distribution is valid. (Give distribution type, etc.) ii. What is the probability that a sample of size 60 gives a sample proportion less than .75 as the proportion who are Vitamin D deficient? d. Suppose a random variable X is Binomial(60,.8). What is the probability X is less than 45? Include any necessary checks for using approximations to distributions.