2.8 Review Questions
- In 2004, the mean net worth of families in the United States was $448.2 thousand and the median net worth was $93.1 thousand. Which measure of center do you think is more appropriate? Explain your answer.
- Wayne Gretzky, a retired professional hockey player, played 20 seasons in the National Hockey League (NHL) from 1980 through 1999. The number of games in which Gretzky played during each of his 20 seasons in the NHL are as follows: 74, 80, 73, 78, 78, 45, 80, 79, 79, 80, 48, 64, 80, 70, 80, 74, 82, 81, 80, 82.
- Find the mean, median, mode of these 20 numbers. Interpret the three measures for the center.
- Find the quartiles of the data and interpret.
- Find the range, interquartile range, sample standard deviation of the data and interpret.
- Find the five-number summary of the data and draw a boxplot of these 20 numbers. Comment on the resulting boxplot.
- Choose proper measures for the center and spread (variation) of the distribution. Justify your answer.
- The following table gives the salaries (in thousand dollars) for physics and computer science (CS) majors obtaining a bachelor’s degree, a master’s degree or a PhD.
- What can we tell from the side-by-side boxplot comparing the salaries of computer science (CS) and physics majors, the one on the left?
- What can we tell from the side-by-side boxplot comparing the salaries of Bachelor, Master and PhD, the one on the right?
- What can we tell from the side-by-side boxplot comparing the salaries of combinations of ”Major” and ”Degree”?
SALARY MAJOR DEGREE 51.9 Physics Bach 58.2 Physics Bach 49.9 Physics Bach 50.6 Physics Bach 51.4 Physics Bach 43.7 Physics Bach 52.9 Physics Bach 59.2 Physics Master 60.5 Physics Master 57.1 Physics Master 59.1 Physics Master 54.9 Physics Master 61.7 Physics Master 62.4 Physics Master 78.2 Physics PhD 69.6 Physics PhD 70.5 Physics PhD 73.2 Physics PhD 81.7 Physics PhD 74.8 Physics PhD 69.8 Physics PhD SALARY MAJOR DEGREE 50.8 CS Bach 59.4 CS Bach 55.9 CS Bach 45.1 CS Bach 54.1 CS Bach 50.7 CS Bach 46.8 CS Bach 65.8 CS Master 57.5 CS Master 66.9 CS Master 62.8 CS Master 68.5 CS Master 69.3 CS Master 61.5 CS Master 73.3 CS PhD 65.7 CS PhD 71.7 CS PhD 72.5 CS PhD 73.0 CS PhD 67.2 CS PhD 67.5 CS PhD
- The z-score corresponding to an observed value of a variable tells you ____________________ .
- A positive z-score indicates that the observation is ______________ the mean, whereas a negative z-score indicates that the observation is ______________ the mean.
- Suppose that you obtained 350 points in an exam. The exam has 400 possible points, the mean score is 280 and the standard deviation is 20. Did you do well on the exam? Explain your answer.
- Each year, thousands of high school students bound for college take the Scholastic Assessment Test (SAT). This test measures the verbal and mathematical abilities of prospective college students. Student scores are reported on a scale that ranges from a low of 200 to a high of 800. In one high school graduating class, the mean SAT math score is 528 with a standard deviation of 105; the mean SAT verbal score is 475 with a standard deviation of 98. A student in the graduating class scored 740 on the SAT math and 715 on the SAT verbal. Compared to the other students in the graduating class, on which test did the student do better?
Show/Hide Answer
- Since the mean is much larger than the median, the distribution is extremely right skewed. It is more appropriate to use the median to describe the center.
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- The sum of the data is [latex]\sum x_i = 1487[/latex] and the sum of squares of the data is [latex]\sum x_i^2 = 112665[/latex]. Therefore, sample mean [latex]\bar x=\frac{\sum x_i}{n}=\frac{1487}{20}=74.35[/latex]. Arrange the data from smallest to largest: 45 48 64 70 73 74 74 78 78 79 79 80 80 80 80 80 80 81 82 82.
Sample size n = 20, median is [latex]\frac{79+79}{2}=79[/latex]. The mode is 80.
Interpretation: 50% of observations are below 70 and another 50% are above 79 (the median); the average of the observation is 74.35 (the mean); the observation occurs most often is 80 (the mode). - The first half: 45 48 64 70 73 74 74 78 78 79, [latex]Q_1=\frac{73+74}{2}=73.5[/latex], Q2 = 79. The second half: 79 80 80 80 80 80 80 81 82 82, [latex]Q_3=\frac{80+80}{2}=80[/latex]. Therefore, the quartiles are Q1 = 73.5, Q2 = 79, Q3 = 80.
Interpretation: the bottom 25% of observations are below 73.5, 25% are between 73.5 and 79, 25% are between 79 and 80, the top 25% are above 80. - Range=Max-min=82-45=37. IQR=Q3 − Q1 = 80 − 73.5 = 6.5.
[latex]s=\sqrt{\frac{\sum x^2-\frac{(\sum x)^2}{n}}{n-1}}=\sqrt{\frac{112665-\frac{1487^2}{20}}{20-1}}=10.5295.[/latex]
Interpretation: the data spread over an interval of length 37 (range); the middle 50% of the observations spread over an interval of length 6.5 (IQR); roughly speaking, the average distance from the observations to the sample mean 74.35 is 10.5295 (standard deviation). - The 5-number summaries are min = 45, Q1 = 73.5, Q2 = 79, Q3, max = 82. The distribution is left-skewed with two outliers at the lower end.
- Use median for the center and IQR for the spread (variation) since outliers exist.
- The sum of the data is [latex]\sum x_i = 1487[/latex] and the sum of squares of the data is [latex]\sum x_i^2 = 112665[/latex]. Therefore, sample mean [latex]\bar x=\frac{\sum x_i}{n}=\frac{1487}{20}=74.35[/latex]. Arrange the data from smallest to largest: 45 48 64 70 73 74 74 78 78 79 79 80 80 80 80 80 80 81 82 82.
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- CS majors have a higher median salary. Physics majors have a larger variation in salary. The distribution for CS majors is left skewed; while the distribution for physic majors is right skewed.
- The median salary for PhD is higher than master, and master is higher than bachelor. Salary for master has a larger IQR than the other two groups. The variations for PhD and bachelor are similar.
- For PhD and bachelor, physics majors have a slightly higher median; for master, however, CS majors have a higher median salary than physics. The variation is similar for CS majors at the three different education level; the variations in salary increase for physics majors when the education level increases.
- how far the observation is away from the mean in units of standard deviation.
- A positive z-score indicates that the observation is above the mean, whereas a negative z-score indicates that the observation is below the mean.
- The z-score is [latex]z=\frac{350-280}{20}=3.5[/latex]. You did extremely well since you are 3.5 standard deviations above the mean. Most x-scores are between -3 and 3.
- The z-score for math is: [latex]z_1=\frac{740-528}{105}=2.02[/latex]. The z-score for verbal is [latex]z_2=\frac{715-475}{98}=2.45[/latex]. The student did better in verbal, since it has a larger z-score.
