AP Statistics Review for Midterm Part 1 (Chapters 1 – 5)

Chapter 1In the paper “Reproduction in Laboratory colonies of Bank Vole,” the authors presented the results of a study of litter size. (A vole is a small rodent with a stout body, blunt nose, and short ears.) As each new litter was born, the number of babies was recorded, and the accompanying results were obtained.

1 4 4 5 5 6 6 7 7 82 4 5 5 5 6 6 7 7 83 4 5 5 6 6 7 7 8 93 4 5 5 6 6 7 7 8 104 4 5 5 6 6 7 7 8 11

The authors also kept track of the color of the first born in each litter. (B = brown, G = gray, W = white, and T = tan)

B B T W T G G G B BW B W B T T G B T BG B B B B G W G T GB B B B G G T T W GB G T W B G T W G W

1. Which variable, litter size or color, is categorical? Color

2. Which variable is quantitative? Litter Size

3. Make a bar chart of the colors.

4. Make a histogram of the litter sizes.

5. Make a dotplot of the litter sizes.

6. Describe the shape of the distribution of litter sizes. Approx. symmetrical

7. Find the mean of the litter sizes. 5.84

8. Is the mean resistant to outliers? No

9. Find the median of the litter sizes. 6

10. Is the median resistant to outliers? Yes

11. Find the 5-number summary of the litter sizes. 1, 5, 6, 7, 11

12. What is the interquartile range? 5 to 7 (or 2)

13. Are there any outliers in the litter sizes (show work using 1.5 IQR)? 1 and 11

14. Make a boxplot and modified box plot (if there are outliers) of the litter sizes.

15. Use the calculator to find the variance of the litter sizes. 3.606

16. Find the standard deviation of the litter sizes, using the calculator. 1.899

17. Is standard deviation resistant to outliers? No

18. Find the degrees of freedom of the litter sizes. 49

19. Make a back-to-back split stemplot of the following Reading Scores:

4th Graders 12 15 18 20 20 22 25 26 28 2931 32 35 35 35 36 37 39 40 42

7th Graders 1 12 15 18 18 20 23 23 24 2527 28 30 30 31 33 33 33 35 36

20. Make a comparison between 4th grade and 7th grade reading scores based on your stemplot.4th graders scored higher than the 7th graders overall. One 7th grader had a score of 1 and there were no 7th grade scores in the 40s. The lowest 4th grade score was 12 with 2 4th grade students scoring in the 40s. Most students in both grades scored in the 30s.

Chapter 2

21. What is the area under any density curve? 1

22. The (mean or median) of a density curve is the equal-areas point, the point that divides the area under the curve in half.

23. The (mean or median) of a density curve is the balance point, at which the curve would balance if made of solid material.

24. If a density curve is skewed to the right, the (mean or median) will be further to the right than the (mean or median).

25. What is the difference between x and µ?x average for a sample (statistic); µ average for a population (parameter)

26. What is the difference between s and σ? s standard deviation for sample (statistic) and σ st dev for a population (parameter)

27. Name 3 properties of normal curves are density curves:Bell shaped, symmetrical, mean = median, follows 68-95-99.7 Rule, almost all the data within ± 3 st dev from the mean

28. How do you find the inflection points on a normal curve?Where the curve changes direction (indicates ±1 standard deviation from the mean)

29. Sketch the graph of N(266, 16), the distribution of pregnancy length from conception to birth for humans.

30. What is the 68-95-99.7 rule?In a normal distribution: 68% of data is within ±1 standard deviation from the mean

95 of data is within ±2 standard deviations from the mean99.7% of data is within ±3 standard deviations from the mean

31. Using the empirical rule (the 68-95-99.7 rule), find the length of the longest 16% of all pregnancies. Sketch and shade a normal curve for this situation.

282 and above

32. Using the empirical rule, find the length of the middle 99.7% of all pregnancies. Sketch and shade.

218 to 314 days

33. Using the empirical rule, find the length of the shortest 2.5% of all pregnancies. Sketch and shade.

234 days and less

34. Using the empirical rule, what percentile rank is a pregnancy of 218 days?

0.15%

Use the information from the distribution of pregnancy length N(266, 16), the Z formula and the Z table to answer questions 33 – 43. Show your work!

35. What percentile rank is a pregnancy of 298 days? 97.5%

36. What percentile is a pregnancy of 250 days? 16%

37. What is the percentile of a pregnancy of 266 days? 50%

38. What z-score does a pregnancy of 279 days have? 0.81

39. What percent of humans have a pregnancy lasting less than 279 days? Sketch and shade a normal curve.

79.1%

40. What z-score does a pregnancy of 257 days have? -0.5625

41. What percent of humans have a pregnancy lasting less than 257 days? Sketch and shade.

0.2877 or 28.77%

42. What percent of humans have a pregnancy lasting longer than 280 days? Sketch and shade.

0.1894 or 18.94%

43. What percent of humans have a pregnancy lasting between 260 and 270 days? Sketch and shade.

0.2467 or 24.67%

44. Would you say pregnancy length is a continuous or discrete variable? Justify.Continuous, it includes a range of values, not specific values.

45. You have normal distributions on your TI-83. Use these functions to check your answers to problems 37 to 41.Use normalcdf(min, max, mean, st. dev) to calculate proportions

46. How long would a pregnancy have to last to be in the longest 10% of all pregnancies?286.48 or 287 days

47. How short would a pregnancy be to be in the shortest 25% of all pregnancies?

255.28 days or 255 days

48. How long would a pregnancy be to be in the middle fifth of all pregnancies?

262 to 270

Chapter 3

49. Graph the following hot dog data:

Calories Sodium (milligrams)108 149130 350132 345135 360138 360140 375144 380145 390150 400163 415167 400172 420176 450180 500184 505195 500200 515

50. What is the explanatory variable?Calories

51. What is the response variable?Sodium

52. Describe the association of the scatterplot (strength, direction, and form)Moderately strong positive, possibly linear relationship between calories and sodium

53. Are there outliers? (Outliers in a scatterplot have large residuals.) (108, 149) doesn’t fit pattern

54. If there are outliers, are they influential? Yes, the correlation changes when this point is eliminated

55. Use your calculator to compute the correlation. 0.9195

56. What percent of the variation of sodium is explained by the LSRL on calories? 0.8455 or 84.55%

57. What two things does correlation tell us about a scatterplot? Strength and direction of a LINE

58. If I change the units on sodium to grams instead of milligrams, what happens to the correlation? Nothing

59. What is the highest correlation possible? 1

60. What is the lowest correlation possible? -1

61. Correlation only applies to what type(s) of relationship(s)? linear

62. Is correlation resistant to outliers? no

63. Does a high correlation indicate a strong cause-effect relationship?

No, only a well-designed experiment can indicate a causation (cause and effect relationship)

64. Sketch a scatterplot with a correlation of about 0.8.

65. Sketch a scatterplot with a correlation of about –0.5.

66. Find the least-squares regression line (LSRL) for the hot dog calories-sodium data.y = - 85.4 + 3.1 x or predicted sodium = - 85.4 + 3.1 (calories)

67. Draw the LSRL on your scatterplot.

68. What is the slope of this line, and what does it tell you in this context?3.1. For every calorie, the sodium increases 3.1 mg

69. What is the y-intercept of this line, and what does it tell you in this context?-85.4 At 0 calories, the sodium in a hot dog would be at -85.4 mg (which doesn’t make sense)

70. Predict the amount of sodium in a hot dog with 155 calories.395.1

71. Predict the amount of sodium in a hot dog with 345 calories.984.1

72. Find the error in prediction (residual) for a hot dog with 180 calories.27.4

73. Find the residual for 195 calories.-19.1

74. Sketch the residual plot of calories vs. residuals and use it to determine if the LSRL is a good model for the data:

75. Why is the prediction for 155 calories acceptable but the prediction for 345 calories not?155 is within the data set, 345 is beyond the data (extrapolation)

76. What point is always on the LSRL? Find this point, and label it on your scatterplot. (x, y) = (156.4, 400.8)

77. Find the standard deviation of the calories.25.6

78. Find the standard deviation of the sodium.86.7

79. Using the equations for slope and y-intercept, verify the slope and intercept of the LSRL.Slope = 3.1 when plug in formula y int = -8.4 when use formulas, so they are close

Chapter 4

80. If you know a scatterplot has a curved shape, how can you decide whether to use a power model or an exponential model to fit data?Exponential has a common ratio (multiplier). Using (x, log y) makes the scatterplot linear.

81. Sketch a graph the following data: Show sketch of scatterplot, it will not be linear

Time (days) Mice30 1960 6090 195120 597

82. Perform the appropriate logarithmic transformation (power or exponential) on the above data to get a transformed linear equation (show a sketch of each scatterplot):

Exponential is the better choice (x, log y)

83. Make a residual plot to support your choice.

84. Transform your best LSRL equation back to exponential or power:Predicted mice = 6.02 (1.039)time

85. Check your equation (from above problem) by using your calculator’s power and exponential regression functionsIt is the same and the correlation matches that of (x, logy)

86. Graph the following data:

Diameter (inches) Cost (dollars)6 3.509 8.0012 14.5015 22.5020 39.50

87. Perform the appropriate logarithmic transformation (power or exponential) on the above data to get a transformed linear equation (show a sketch of each scatterplot):

(logx, logy) or power is best88. Make a residual plot to support your choice.

Residuals still show a pattern, but the correlation is very high

89. Transform your best LSRL equation back to exponential or power:

Predicted cost = (0.095)(x2.02 )

90. Check your equation (from above problem) by using your calculator’s power and exponential regression functions0.095 (x2.017) the equations are the same (with rounding)

91. What is extrapolation, and why shouldn’t we trust predictions using extrapolation?Predicting outside the set of data. Can’t predict “in the future” we only know how variables behave within the data set.

92. What is interpolation, are these predictions acceptable?Predicting within the set of data, this is expected

93. What is a lurking variable?A hidden variable that may influence outcomes

94. Why should we avoid using averaged data for regression and correlation?Averages remove some of the variability (change) and make it appear more consistent than it actually is.

95. What is causation? Give an example.A change in the explanatory variable (x) causes a change in the response variable (y)

EX: Lack of eating causes hunger96. What is common response? Give an example.

Both the explanatory and response variables are changed by a lurking variable (x and y are not related)EX: Reading level and shoe size are not related, both are influenced by age

97. What is confounding? Give an example.The explanatory variable may cause changes in the response variable, but there are other variables (lurking) that are also influencing the response variable. We cannot separate the influence of x and the other variable(s) on y.

EX: Parent’s income may have an influence on their children’s income, but so do many other things such as jobs, hours worked, motivation, etc.

98. What type of data do we put in a two-way table?Categorical

Use this table for the next few questions:

Smoking StatusEducation Never smoked Smoked, but quit Smokes Total Did not complete high school 82 19 113 214 25.9%Completed high school 97 25 103 225 27.3%1 to 3 years of college 92 49 59 200 24.2%4 or more years of college 86 63 37 186 22.5%

Total: 357 (43.3%) 156 (18.9%) 312 (37.8%) 825

99. Fill in the marginal totals in the table above.

100. Fill in the marginal distributions in the table above

101. What percent of these people smoke?37.8%

102. What percent of never-smokers completed high school?27.2%

103. What percent of those with 4 or more years of college have quit smoking?33.9%

104. What percent of those with some college smoke?29.5%

105. What percent of smokers did not finish high school?36.2%

106. Create segmented bar graphs for each smoking status

107. Are smoking status and education independent?No the more education people receive, the less likely they are to smoke

108. What is Simpson’s Paradox?A contradiction in results when data is combined and when it is separated (indicates the presence of a lurking variable)

109. What do you look for in a data table that might “suggest” a Simpson’s Paradox is present?Look for a category with a significant different number in the total for that category. For example, if most of the totals are 500 or more and one total is below 100, there could be a paradox.

Chapter 5110. What is the difference between an observational study and an experiment?

Observation study just gathers results (passive), an experiment actual imposes a treatment (active)

111. What is a voluntary response sample?People choose whether or not to participate (they choose themselves instead of being chosen)

112. How are a population and a sample related but different?A population includes ALL and a sample includes PART. A good, unbiased sample will represent the population well.

113. Why is convenience sampling biased?When you ask friends, family, or those who are easy to ask, they won’t represent the population…. Just people that are similar to you.

114. SRS stands for what kind of sample? Name and define.Simple Random Sample. Everyone has the same chance of being selected as every one else.

115. Discuss how to choose a SRS of 4 towns from this list:

Allendale Bangor Chelsea Detour Edmonton FennvilleGratiot Hillsdale Ionia Joliet Kentwood Ludington

Number the towns 1 – 12 if using a calculator or 01 – 12 if using a random number table. In calculator use the random integer function RandInt(1,12) to select 4 unique numbers (ignore repeats). In the random number table choose 4 two digit numbers (ignore any that are more than 12). Find the towns that correspond to the four unique numbers that are chosen.

116. What is a stratified random sample?Separate into groups by common traits and then choose a SRS from each group.

117. What is a multistage sample?Separate into groups by common traits, then randomly choose some of those groups. Separate the groups you chose into smaller groups and randomly choose some of those, etc.

118. What is undercoverage?Not enough people are asked.

119. What is nonresponse?People are asked, but they do not answer

120. What is response bias?Answers do not reflect the actual beliefs (ex: lie, don’t understand the questions)

121. Why is the wording of questions important? Give an example.Words can be misleading and favor an option.EX: Do you believe it is ok to destroy the planet by not recycling?

122. How are experimental units and subjects similar but different?Units are what you experiment on unless they are people, then they are subjects.

123. What is the placebo effect?Reacting to a treatment because you believe that you have received it

124. What is the purpose of a control group?Used for comparison (nothing else should be different between the treatment group and the control group except the actual treatment received).

125. Give an example of when we may not want to use a placebo/control group.When it is not ethical to do so.Brain surgery, you wouldn’t want to do the procedure as a “fake”

126. What are matched pairs experiments? Give an example.Each subject receives both treatments.EX: left hand and right hand strength or a pre-test and post-test.When you want to compare a person to themselves.

127. What are the components of a well-designed experiment?Randomization, Control, Comparison, Replication

128. What does double-blind mean, and why would we want an experiment to be double-blind?Neither the subjects nor the evaluator is aware of what treatment was given. This should be done when the evaluator may have bias (such as financial interest) in proving something works.

129. What is block design and how is it different than stratified random samples?Block design separates units into groups with common traits before the experiment is started. Then experiments are run on each group separately.Blocking is done in experiments, stratified samples are done in surveys.

130. I want to test the effects of aerobic exercise on resting heart rate. I want to test two different levels of exercise, 30 minutes 3 times per week and 30 minutes 5 times per week. I have a group of 20 people to test, 10 men and 10 women. I will take heart rates before and after the experiment. Draw a diagram for this experimental design.

131. Why is simulation useful?Saves time and money

132. What are the five steps of a simulation?State the problem, state assumptions, assign digits to represent outcomes, run simulation, conclude

133. Design and perform a simulation of how many children a couple must have to get two sons. (A simulation involves many trials. For this simulation, perform 10 trials.). Show your results below:Find the number of children a couple must have to get 2 sons. Assume random and independence.Let 0 = Girl and 1 = Boy (or flip a coin).Results will vary (show yours) and then based on your 10 trials, give the average number of children a couple had in order to get 2 sons.

134. If John makes 70% of his free throw shots, use your calculator to simulate how many shots he will make in 10 trials. Do this simulation 5 times and show your results below:Determine how many shots John makes in 10 trials. Assume these are independent and he makes 70% of these.Let 1 – 7 = Makes the shot and 8 – 9 and 0 = Misses the shot. Look at 10 digits (in calculator and on table) and count how many he makes, do this 5 times.Results will vary (show yours) and then based on your 5 simulations, find the average.