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Chapter 4: Problem 46

Suppose that your statistics professor returned your first midterm exam with only a \(z\) score written on it. She also told you that a histogram of the scores was approximately normal. How would you interpret each of the following \(z\) scores? a. \(2.2\) d. \(1.0\) b. \(0.4\) e. 0 c. \(1.8\)

Short Answer

Expert verified
A z-score indicates how many standard deviations a data point is from the mean. Z-scores of 2.2, 1.8, 1.0, 0.4 and 0 place the data point in the 98th, 96th, 84th, 66th and 50th percentiles respectively.

Step by step solution

01

Interpret the score of 2.2

A z-score of 2.2 indicates that the score is 2.2 standard deviations above the mean. In terms of percentile, this score is approximately in the 98th percentile, which means only about 2% of the class scored higher.
02

Interpret the score of 1.0

A z-score of 1.0 indicates that the score is one standard deviation above the mean. In terms of percentile, this score is in the 84th percentile, which means that about 84% of the class scored lower and 16% scored higher.
03

Interpret the score of 0.4

A z-score of 0.4 means the score was 0.4 standard deviations above the mean. In terms of percentile, this score is approximately in the 66th percentile, meaning about 66% of the class scored lower and 34% scored higher.
04

Interpret the score of 0

A z-score of 0 means the score is the same as the mean. Therefore, half the students scored lower than this and half scored higher, placing this score in the 50th percentile.
05

Interpret the score of 1.8

A z-score of 1.8 means the score is 1.8 standard deviations above the mean. In terms of percentile, this score is approximately in the 96th percentile, meaning about 96% of the class scored lower and 4% scored higher.

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Most popular questions from this chapter

The paper "Answer Changing on Multiple-Choice Tests" (Journal of Experimental Education [1980]: 18-21) reported that for a group of 162 college students, the average number of responses changed from the correct answer to an incorrect answer on a test containing 80 multiplechoice items was 1.4. The corresponding standard deviation was reported to be 1.5. Based on this mean and standard deviation, what can you tell about the shape of the distribution of the variable number of answers changed from right to wrong? What can you say about the number of students who changed at least six answers from correct to incorrect?

The Highway Loss Data Institute reported the following repair costs resulting from crash tests conducted in October 2002 . The given data are for a 5 -mph crash into a flat surface for both a sample of 10 moderately priced midsize cars and a sample of 14 inexpensive midsize cars. \(\begin{array}{lrrrrr}\text { Moderately Priced } & 296 & 0 & 1085 & 148 & 1065 \\ \text { Midsize Cars } & 0 & 0 & 341 & 184 & 370 \\ \text { Inexpensive } & 513 & 719 & 364 & 295 & 305 \\ \text { Midsize Cars } & 335 & 353 & 156 & 209 & 288 \\ & 0 & 0 & 397 & 243 & \end{array}\) a. Compute the standard deviation and the interquartile range for the repair cost of the moderately priced midsize cars. b. Compute the standard deviation and the interquartile range for the repair cost of the inexpensive midsize cars. c. Is there more variability in the repair cost for the moderately priced cars or for the inexpensive midsize cars? Justify your choice. d. Compute the mean repair cost for each of the two types of cars. e. Write a few sentences comparing repair cost for moderately priced and inexpensive midsize cars. Be sure to include information about both center and variability.

The average reading speed of students completing a speed-reading course is 450 words per minute (wpm). If the standard deviation is \(70 \mathrm{wpm}\), find the \(z\) score associated with each of the following reading speeds. a. \(320 \mathrm{wpm}\) c. \(420 \mathrm{wpm}\) b. \(475 \mathrm{wpm}\) d. \(610 \mathrm{wpm}\)

Suppose that 10 patients with meningitis received treatment with large doses of penicillin. Three days later, temperatures were recorded, and the treatment was considered successful if there had been a reduction in a patient's temperature. Denoting success by \(S\) and failure by \(\mathrm{F}\), the 10 observations are \(\begin{array}{llllllllll}\mathrm{S} & \mathrm{S} & \mathrm{F} & \mathrm{S} & \mathrm{S} & \mathrm{S} & \mathrm{F} & \mathrm{F} & \mathrm{S} & \mathrm{S}\end{array}\) a. What is the value of the sample proportion of successes? b. Replace each \(\mathrm{S}\) with a 1 and each \(\mathrm{F}\) with a 0 . Then calculate \(\bar{x}\) for this numerically coded sample. How does \(\bar{x}\) compare to \(p ?\) c. Suppose that it is decided to include 15 more patients in the study. How many of these would have to be S's to give \(p=.80\) for the entire sample of 25 patients?

In 1997 a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessey v. Digital Equipment Corporation). The jury awarded about \(\$ 3.5\) million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a "normative" group of 27 similar cases and specified a reasonable award as one within 2 standard deviations of the mean of the awards in the 27 cases. The 27 award amounts were (in thousands of dollars) \(\begin{array}{rrrrrrrr}37 & 60 & 75 & 115 & 135 & 140 & 149 & 150 \\ 238 & 290 & 340 & 410 & 600 & 750 & 750 & 750 \\\ 1050 & 1100 & 1139 & 1150 & 1200 & 1200 & 1250 & 1576 \\ 1700 & 1825 & 2000 & & & & & \end{array}\) What is the maximum possible amount that could be awarded under the "2-standard deviations rule"?
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