mid2soln - Midterm Two Solutions General Comments The...

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Midterm Two Solutions General Comments: The average on this exam was 82.3, a little lower than midterm 1 (84.5) but not much. The median was 83.5, low score 36, high score 100 (wow!) and standard deviation 11.4. I have given approximate grade distributions below. As always remember that they do not necessarily represent the score necessary for a particular grade overall in the course as you get to drop your lowest homework and drop the weight on your lower midterm to 10% if you do better on the final. These are mostly designed to give you an idea of your relative performance on this exam. People struggled a lot on the last question–partly this may have been time, but historically people have difficulties when I give a problem like this so make sure you read the solutions carefully. In addition I recommend you look carefully at the solutions to the second half of problem 2, especially the part about (g) about multicollinearity and overfitting, and the parts on interactions. There seem to be some common misconceptions about these concepts. Grade Score Number In Range A 89+ 30 A- 84-88.5 19 B+ 80-83.5 18 B 70-79.5 20 B- 65-69.5 3 C+ 60-64.5 3 C 55-59.5 3 C- 50-54.5 1 F ¡ 50 2 THE STORY BEHIND THE EXAM: As we all know from class my baby has not been sleep- ing well...perhaps that has something to do with the exam theme! A professor of pediatrics at the University of Calculationally Learned Adults is interested in knowing what factors affect how well an infant sleeps at night. During the course of the exam you will help her analyze the data from a study she has conducted. Question 1: Drinking the Milk of Paradise (45 points, 45 minutes) A common story among parents of young babies is that if you can get them to drink a lot of milk before going to bed they will sleep longer. Our pediatric researcher decides to test this. She has collected information on Y, the number of minutes a baby sleeps and X, the number of ounces of 1
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milk the baby has consumed. Some summary statistics, a simple linear regression printout and sev- eral corresponding plots are shown below. Use them to answer the questions on the following pages. ¯ Y = 562 ¯ X = 4 . 75 SSX = 755 n = 100 . reg Sleep Milk Source | SS df MS Number of obs = 100 -------------+------------------------------ F( 1, 98) = 28.98 Model | 1034501.63 1 1034501.63 Prob > F = 0.0000 Residual | 3497999.72 98 35693.8747 R-squared = 0.2282 -------------+------------------------------ Adj R-squared = 0.2204 Total | 4532501.35 99 45782.8419 Root MSE = 188.93 ------------------------------------------------------------------------------ Sleep | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- Milk | 37.02467 6.877371 5.38 0.000 23.37675 50.67259 _cons | 385.8047 37.77206 10.21 0.000 310.8473 460.7621 ------------------------------------------------------------------------------ Part a (3 points) Is there a significant linear relationship between sleeping time and amount of milk drunk? Justify your answer with an appropriate test using α = . 05. You do not need to write out all the details. Just give your basic reasoning. Solution: Yes, the p-value for the F test (or the t-test for the Milk variable) is 0.0000 which is certainly much less than α = . 05 so we reject the null hypothesis that β 1 = 0 and conclude that there is a significant relationship between amount of milk drunk and length of time slept.
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