Buying a Home. Suppose that you are thinking of buying a resale home in a large tract. The owner is asking $205,500. Your realtor obtains the sale prices of comparable homes in the area that have sold recently. The mean of the prices is $220,258 and the standard deviation is $5,237. Does it appear that the home you are contemplating buying is a bargain? Explain your answer using the z-score and Chebyshev’s rule. Comparing Relative Standing. If two distributions have the same shape or, more generally, if they differ only by center and variation, then z-scores can be used to compare the relative standings of two observations from those distributions. The two observations can be of the same variable from different populations or they can be of different variables from the same population. Consider Exercise. Exercise SAT Scores. 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. Summary results for the scores are published by the College Entrance Examination Board in College Bound Seniors. 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. a. Under what conditions would it be reasonable to use z-scores to compare the standings of the student on the two tests relative to the other students in the graduating class?b. Assuming that a comparison using z-scores is legitimate, relative to the other students in the graduating class, on which test did the student do better?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Buying a Home. Suppose that you are thinking of buying a resale home in a large tract. The owner is asking $205,500. Your realtor obtains the sale prices of comparable homes in the area that have sold recently. The mean of the prices is $220,258 and the standard deviation is $5,237. Does it appear that the home you are contemplating buying is a bargain? Explain your answer using the z-score and Chebyshev’s rule.
Comparing Relative Standing. If two distributions have the same shape or, more generally, if they differ only by center and variation, then z-scores can be used to compare the relative standings of two observations from those distributions. The two observations can be of the same variable from different populations or they can be of different variables from the same population. Consider Exercise.
Exercise
SAT Scores. 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
a. Under what conditions would it be reasonable to use z-scores to compare the standings of the student on the two tests relative to the other students in the graduating class?
b. Assuming that a comparison using z-scores is legitimate, relative to the other students in the graduating class, on which test did the student do better?
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