171
Prelim#2
Spring Math 06
Directions: You may use two pages (two sides of a sheet of paper) of notes. Calculators are permitted and we also provide you with tables. All work must be shown in the answer booklet. Be sure it is complete, neat, and in order. Clearly indicate your final answers. Proper mathematical justification must be provided in order to earn full credit; correct answers with no work shown may receive no credit. Note that using the calculator does not take the place of showing your work. You may use standard notation; any new notation or abbreviations you use must be clearly defined. You have 90 minutes to complete this exam. Good luck! 1. [17 points] Hitchcock. The following table lists the running times (in minutes) of Alfred Hitchcock's movies:
Film The Birds Dial M for Murder Family Plot Foreign Correspondent Frenzy I Confess The Man Who Knew Too Much Marnie North by Northwest Notorious The Paradine Case Psycho Rear Window Rebecca Rope Shadow of a Doubt Spellbound Strangers on a Train To Catch a Thief Topaz Under Capricorn Vertigo Running Time 119 105 120 120 116 108 120 130 136 103 116 108 113 132 81 108 111 101 103 126 117 128
a) Display these data in a stemplot. [3] b) Find the 5-number summary of these data. [6] c) Identify any outliers in the data, clearly explaining why you decided they were outliers. [4] d) Display these data with a boxplot. [4]
2. [15 points] Bones. Anthropologists examining human skeletal remains often have only partial skeletons to work with. This makes it impossible to get any direct measurement of the individuals height, but reliable estimates can be generated from measurements of small bones. For example, data the anthropologists collected from 9 complete skeletons found a correlation of 0.856 between the length of the metacarpal bone (in cm) and the height of the individual (in cm). Here are the scatterplot, residuals plot, and regression analysis:
185.0 180.0 175.0 H e i g h t 170.0 165.0 160.0
7.5 r e s i d u a l s ( H / M ) 5.0 2.5 0.0 -2.5
39
42
45
48
160
165
170
175
180
MetaCarp
predicted(H/M)
Dependent variable is: Height 9 total cases s = 4.255 with 9 - 2 = 7 degrees of freedom Source Regression Residual Variable Constant MetaCarp Sum of Squares 347.291 126.709 Coefficient 94.4279 1.69963 df 1 7 Mean Square 347.291 18.1013 F-ratio 19.2
s.e. of Coeff 17.69 0.3880
t-ratio prob 5.34 0.0011 4.38 0.0032
a) Explain briefly why linear regression appears to be appropriate for these data. [2] b) Write the equation of the least squares regression line for predicting height from the length of the metacarpal bone. [4] c) Interpret the slope of the line in this context. [3] d) Using this model, what fraction of the variability in height can be explained by differences in the length of the metacarpal bones? [2] e) One individual whose metacarpal bone measured 42 cm was later determined to be 161 cm tall. Find the residual. [4] 3. [11 points] Jellybeans. Suppose that 20% of all jellybeans from a certain candy company are black. You buy a bag of 120 assorted jellybeans. a) What is the probability that less than 15% of the jellybeans in your bag will be black? [7] b) Discuss the assumptions and conditions necessary to justify your answer. [4] 4. [11 points] Carpets. Rugs-R-Us manufactures low-cost carpets for college dorm rooms. Past inspections indicated that their carpets have an average of 1.6 flaws, with a standard deviation of 1.2 flaws. The Campus Store orders 200 of these carpets, asking Rugs-R-Us to supply a random assortment of patterns and colors. a) Find the approximate probability that the mean number of flaws for the carpets this in order is more than 1.8 per carpet. [7] b) Name a theorem you had to use to answer this question, and briefly explain how it was important in getting your answer. [4]
5. [19 points] Video games. A psychology researcher wondered if people who play video games have better spatial perception than those who dont. She distributed a questionnaire to all the students in a local high school, asking them how many hours a week they spent playing video games. Based on the students answers, she identified two groups of teenagers: those who never played video games and those who played over 10 hours per week. The researcher then selected 25 students at random from each group and tested each of them for spatial reasoning ability. a) Is this an experiment or an observational study? Explain briefly. [2] b) Write a sentence that represents her null hypothesis. (Express it in words - dont try to use statistical notation.) [2] c) Write a sentence that represents her alternative hypothesis. [2] d) When she analyzed the test results she found that the frequent video game players had a higher mean spatial reasoning score than those who never played. She reported that the results were statistically significant. Explain what that means in this context. [2] e) If further research demonstrates that her preliminary conclusion was wrong, which type of error did she commit? [2] f) Upon hearing of the results of the initial research, a newspaper ran a story with the headline Playing video games improves spatial ability. Briefly explain why this headline is misleading and offer an alternative explanation for the results. [3] g) Briefly describe the design for an experiment that could justify the newspapers conclusion. [6] 6. [7 points] Flipping a coin. In a 2004 paper, professional magician turned statistician Persi Diaconis (and associates) analyzed the mechanics and physics of coin flipping. From their models they concluded that a coin flipped in the air and caught by hand is not quite fair. In particular, there appears to be a slight bias toward the coin landing with the same face upward as when it was launched. The mathematical models predict this outcome with probability p...