Honor Code statement on the back 100 pts. Maximum of 75 minutes. of the last page.

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1 No Name on This Page!!! Biostatistics Please write your name and the Examination #3 Honor Code statement on the back 100 pts. Maximum of 75 minutes. of the last page. Many professors within the sciences at Cedar Crest have active research programs dealing with a great diversity of organisms [e.g., flowers, stinkbugs, leeches, fruit flies, fish, fungi, bacteria, viruses, and criminals]. Let s take a look at some experimental questions related to current research at CCC. On the next three pages you will find research scenarios and associated data sets. Your mission is to conduct the analyses and form the final conclusions. A word of advice: Make sure you understand each project before you begin to analyze the data. #1 (25 pts.) In class we learned that Dr. Reese and her students are interested in Cryptococcus. It turns out that she and her research students have other interests as well. Not long ago one of her students shared her work with her peers and members of the science faculty. Ms. J. has been spending long hours in OBC2 identifying bacterial species found within the mouths of students. Deviating a bit from her actual study, let s say that they are interested in determining whether the mean richness of the bacterial flora within the mouths of students at colleges within the Lehigh Valley is different than that found within the mouths of criminals housed in the local prison. Richness is defined as the number of different species present. Each value in the table below indicates the richness value (i.e., # of spp.) for an individual within one of the two groups. Sample size = 20 for each group. In the space below the table and on the next page, identify the specific test Dr. Reese and her students should use, state the hypotheses using proper notation, show the values that help them decide on their conclusion, and state the conclusion (statistical format and sentence format). College Students Prison Criminals The Two Sample t-test is the best statistical test for this situation because two samples are being compared. There is no justification for performing a paired t-test unless all of the college students are planning to spend some years in prison following graduation. This would be a good before & after study, but I m certainly not recommending it. In the description it is clear that the researchers just want to find out if there is a difference between the two populations. This means that they will be running a twotail test. Consequently, H 0 : µ college = µ prison and H A : µ college µ prison Is there any reason to assume that the two samples have equal variances? Not really. This will be supported when you see their actual variances. Consequently, one should run Data Analysis/t-Test Two Sample Assuming Unequal Variances

2 2 The outcome of the analysis is therefore, reject the null because P= This means that there is a significant difference between college students and criminals, at least in terms of the richness of bacteria on their teeth. Judging by the means, college students have fewer types of bacteria on their teeth than do the criminals. #2 (25 pts.) One of the questions being addressed in Dr. Faivre s lab deals with the infertility of a plant species found in Florida. There is concern about local extinction of this species, and one factor that may lead to this extinction is the plant s suspected inability to reproduce as easily as populations in other regions of the country. Dr. Faivre has a lot of experience with fluorescent microscopy and the use of this tool to monitor the formation of pollen tubes, which are necessary for the fertilization of eggs within the ovary of a flower. In theory, every pollen grain (from the same species) that falls on the stigma of a flower has the potential to form a pollen tube and ultimately provide a path to an egg for a sperm. For the purposes of this exam, let s say based on information within the primary literature, that within viable populations of this plant species the average percentage of the pollen tube attempts that make it to the ovary of the flower is 85%. Dr. Faivre happens to believe that her relatively infertile plant population will exhibit a significantly lower success rate (%) in reaching the ovary. She and her students decide to spend Spring Break in Florida. The product of their hard work is in the table below. The values in the table below indicate how many pollen tubes begin to grow in each plant and how many of those pollen tubes grow until they reach the planned destination (ovary). Sample size = 20; each column includes data from the same flower. In the space below the table, identify the specific test Dr. Faivre and her students should use, state the hypotheses in proper format, show the values that help them decide on their conclusion, and state the conclusion (statistical format and sentence format).

3 3 # of Pollen Tube Attempts # of Completed Pollen Tubes The One Sample t-test is the best statistical test for this situation because they are interested in comparing their sample with an average value found in the literature. Raw data from the published study are not available. Recall that they believe that the relatively infertile population will exhibit a lower percentage rate. This defines the alternative hypothesis. Consequently, H 0 : µ Florida 85 and H A : µ Florida < 85 Sample size (n) is less than 30 so it s important to use the t-statistic. The first step is to calculate the t-statistic. Recall that t calc = [sample mean - µ Ho ] / standard error. Running Data Analysis/Descriptive Statistics we find that the sample mean is and the standard error is Given that the µ defined in the null hypothesis equals 85, our calculated t-statistic turns out to be: So far, so good; it s negative which means the Florida population shows signs of trouble with their pollen tubes. But, we need to find out if the difference is significant. A comparison between the calculated t-statistic and the critical t-statistic give us an objective answer. Excel will deliver. Recall: =TINV(probability,degrees of freedom) will give you the critical t-value. Don t forget that we need to kick the probability value up to 0.1 from 0.05 because we re doing a one-tail test (see hypotheses). Degrees of freedom in the case of this study is 20-1=19. Excel gives us a critical t-value = Recall that our calculated t-value = t-value just means that we re focusing on the critical value to the calculated t-value is between the crit ical t-value and the mean, which means we does not show a significant difference between their population and other ulations, at least in terms of the success rate of pollen tube formatio #3 (25 pts.) Professor Ritter and a member of her research lab are interested in the effect of time on the detection of male genetic material in rape cases. They happen to believe that male epithelial cells have greater longevity within the vaginal cavity than do male spermatozoa cells. Suppose they set up an experiment to test this belief and end up with the following data set. The values in the table indicate how long (number of hours) the specific cells survive within the vaginal cavity. Sample size = 10. Ten women were involved in the study, and the two values within each set of readings (epithelial cells and

4 4 spermatozoa) are from the same woman. It s time to analyze the data. Be sure to choose the statistical test that has the greatest statistical power (1-β). In the space below the table, identify the specific test Professor Ritter and her students should use, state the hypotheses in proper notation, show the values that help them decide on their conclusion, and state the conclusion (statistical format and sentence format). Epithelial Cells Spermatozoa The Paired t-test is the best statistical test for this situation because two readings are available for each individual. If they were to use a standard two-sample t-test, the error associated with differences among individuals would become a factor in the analysis. Recall that they believe that epithelial cells will last longer than spermatozoa. This defines the alternative hypothesis. Consequently, H 0 : µ epi µ spe and H A : µ epi > µ spe t-test: Paired Two Sample for Means esence of less than or equal and greater than signs in the hypotheses, a one-tail test is performed. The calculated probability for a one-tail test = Recall, if P 0.05, then reject the null hypothesis. The outcome of the analysis is therefore, reject the null. This means that epithelial cells do in fact last longer than spermatozoa cells in the vaginal cavity.

5 5 #4 (25 pts.) It s time to hug a tree; so many to choose from on the Cedar Crest campus! Suppose you were asked to go out into nature to measure the circumference of 30 trees on campus. Other members of the class were asked to do the same, each with n=30. In all, 16 sets of data would be collected using proper sampling techniques. After hours of hard labor, raw data came pouring in and the mean (in cm) of each data set was computed and placed in the following table. Student Mean Student Mean a. (5 pts.) Explain how you would properly sample the trees on campus to yield a data set worthy of statistical analyses. A number of approaches to random sampling exist (e.g., simple random sampling; systematic sampling; cluster sampling; stratified sampling). One approach would be to use a GPS unit to define the position of each tree on campus and then use GIS software to map the placement of each tree. After assigning an ID number to each tree, randomly select 30 trees. Then all you would do is go out and measure the circumference of each of the 30 trees. b. (10 pts.) Draw a histogram using the mean values. Label the axes and explain what specific distribution you re looking at (I m not looking for a specific probability distribution). You re looking at a distribution of means.

6 6 c. (5 pts.) Suppose you used the raw data (n=480) to create a histogram. How would the histogram look as compared to the histogram in the previous question (no need to draw, just explain)? The histogram of individual l observations would include a much greater range of values. In addition, even if the distrib ution of individual observations is not normally distributed, the distribution of means is likely to conform to a normal distribution (Central Limit Theorem). d. (5 pts.) Clearly explain why the distribution created in question 4b is so important in the field of statistics. When we compare, for example, one sample with another, we are trying to determine whether the two samples come from the same population, or not. To do this we need to determine if their means are sufficiently far apart to conclude that it is very unlikely that they come from the same population. Sufficiently far apart depends on the distributions of the means and the relative placement of the specific means under consideration. Since we are comparing means, we need to look at the distribution of means (and associated probabilities); the distribution of observations just doesn t cut it.