Answer the following questions and read instructions carefully. Ask any questions please.
Reminders Never blindly punch numbers into a calculator Remember to never ever just punch values into a calculator or statistical software. It's great to have figured out the appropriate formulas to use, and which numbers to enter into them, but never forget that the whole point of statistical analysis is to answer real-world questions. So if your answers do not make "real world" sense – then they are not useful. Providing numeric answers can, of course, be highly misleading and cause businesses, governments, etc. to fail. As an example, you will need to use R’s pnorm() and qnorm() functions for various problems in this assignment. As indicated above, however, be sure that you remind yourself of exactly what these functions tell us. Again, do not ever simply type values into R (or any software) without being clear in your own mind of what a given function does. Academic Integrity Please be aware that you do have resources to help you learn the material. As discussed on the syllabus, there is a free tutoring service available. Also, I am always happy to make an appointment to meet with you. BUT.... though I do want you to do well, I take plagiarism and other forms of cheating extremely seriously. There are several ways of testing to see if students are submitting work that was not their own. It often does get caught eventually. So please do not be one of those students! I feel absolutely terrible when I have to submit an Academic Integrity complaint to the committee and give a student an F in the course, and place a note in their academic file. But I will do so when it happens. The reason is that it is very important to me that all students have the same, fair opportunity to receive credit for the work that they did. Problem #1 A Intro to Psychology class of 117 students took an exam. The test had a maximum score of 70. The results were approximately normally distributed. N (61.4, 4.8) For your calculations, use two decimal places. You can round the second decimal place. For example, 6.886 can be rounded to 6.89. 1. In your own words, describe what is meant by standard deviation. Note: I do not want a mathematical explanation. Rather, I want you to explain in words what the standard deviation is. You can include numbers if you need them to help with your discussion, but I do not want to see the actual formula for SD. This question is worth up to 3 points. You will be given full points based on your ability to convince the grader that you do indeed understand the concept of SD. 2. One student had a score of 66. What percentile does this correspond to? Hint: Remember that R’s pnorm() function gives the area under the density curve. 3. What exam result would be needed to be at the 90th percentile? 4. What percentage of students scored between 55 and 65? 5. What percentage of test takers would have a z-score of lower than -2? 6. What percentage of test takers would have a z-score of greater than +2? Problem #2 Shown here are a series of test scores that were approximately normal, N (15,8). The lowest possible score on the test was 0, and the highest was 25. 16.1, 25, 22.1, 11.7, 4.4, 15.8, 7.5, 10.6, 10.5, 19.9, 22.3, 14.2, 37.8, 12.3, 19.8, 7.5, 7.5, 12.4, 18.7, 5.1 Using R (as well as your mind!), answer the following: 1. Describe the distribution (remember to use the assignment checklist to make sure you can recall what is involved in this step). 2. Draw a boxplot. What does the open circle represent? (Hint: It rhymes with "shmoutlier"). What do you want to do with this observation? Remember that it is always important to justify any decision you make regarding including or excluding certain unusual observations from an analysis. 3. Using a different variable name, generate a second vector without the outlier. (Yes, I am aware that this answers one of the questions from above). Recreate the boxplot. Generate another series of descriptive statistics. 4. Take a look at the histogram. Do you think this variable normally distributed? Let's assume that we think it is. There is another graph we can use to help confirm that the variable is normally distributed. Plot this graph and indicate whether or not the graph supports our theory of normality. Problem #3 You must use the empirical rule to answer the problems in this question. In other words, I do not want you to use R’s pnorm () function to answer any of these questions. Data from a series of average rainfalls over April covering many consecutive years was approximately normally distributed and had a mean of 2.8 inches with a standard deviation of 0.6. 1. One month had a rainfall of 2.2 inches. What percentile does this equate to? 2. What percentage of months had rainfalls between 1.6 and 4.6 inches? 3. What percentage of months had rainfalls above 3.4 inches? 4. What would be the rainfall amount for a month with a z score of -3? 5. Do not use the empirical rule for this problem. That is, you may use R for this: One rainfall was in the 28th percentile. What amount does that correspond to? Hint: There is an R function that will give you this exact piece of information. That is, it converts the area under the normal density curve back into a z-score. Problem #4 Here is a dataset in which the time to complete a swim race was measured along with he resulting pulse (heart rates) at the end of the race. Please note that these are not identical to the data set seen in lecture. There are 8 observations: Create a scatterplot in which the explanatory variable is the swim time (in seconds) and the response variable is the heart rate (in beats per second). You must have all of your R code saved in a script called: swim_race.R For your convenience, I am providing you with the vectors to create the twovariables. You can paste these into your R script: swimTime <- c(33.64,="" 34.23,="" 34.55,="" 33.18,="" 34.29,="" 34.32,="" 33.89,="" 33.9)="" heartrate=""><- c(131.6, 141.9, 147.6, 123.6, 143, 143.5, 136, 136.2) when you are done, your graph should look something like this: c(131.6,="" 141.9,="" 147.6,="" 123.6,="" 143,="" 143.5,="" 136,="" 136.2)="" when="" you="" are="" done,="" your="" graph="" should="" look="" something="" like="">