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Chapter 4: Problem 48

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
Z-scores of 2.2, 1.0, 0.4, 0, and 1.8, respectively, indicate performances that are 2.2, 1.0, 0.4 standard deviations above average, exactly average, and 1.8 standard deviations above average.

Step by step solution

01

Interpretation of a z-score of 2.2

A z-score of 2.2 suggests that the exam score is 2.2 standard deviations above the mean score. This is a high z-score which indicates a good performance in the exam, better than many of the students.
02

Interpretation of a z-score of 1.0

A z-score of 1.0 suggests the exam score is 1 standard deviation above the mean score. This means the score is above average, but not extremely high.
03

Interpretation of a z-score of 0.4

A z-score of 0.4 suggests the exam score is 0.4 standard deviations above the mean score. While the score is above average, it is not by much, indicating a near average performance.
04

Interpretation of a z-score of 0

A z-score of 0 means that the score is exactly the mean score. This suggests an average performance on the exam.
05

Interpretation of a z-score of 1.8

A z-score of 1.8 suggests that the score is 1.8 standard deviations above the mean score. This suggests a well above average performance on the exam.

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Most popular questions from this chapter

The standard deviation alone does not measure relative variation. For example, a standard deviation of \(\$ 1\) would be considered large if it is describing the variability from store to store in the price of an ice cube tray. On the other hand, a standard deviation of \(\$ 1\) would be considered small if it is describing store-to-store variability in the price of a particular brand of freezer. A quantity designed to give a relative measure of variability is the coefficient of variation. Denoted by CV, the coefficient of variation expresses the standard deviation as a percentage of the mean. It is defined by the formula \(C V=100\left(\frac{s}{\bar{x}}\right)\). Consider two samples. Sample 1 gives the actual weight (in ounces) of the contents of cans of pet food labeled as having a net weight of 8 ounces. Sample 2 gives the actual weight (in pounds) of the contents of bags of dry pet food labeled as having a net weight of 50 pounds. The weights for the two samples are \(\begin{array}{lrrrrr}\text { Sample 1 } & 8.3 & 7.1 & 7.6 & 8.1 & 7.6 \\ & 8.3 & 8.2 & 7.7 & 7.7 & 7.5 \\ \text { Sample 2 } & 52.3 & 50.6 & 52.1 & 48.4 & 48.8 \\ & 47.0 & 50.4 & 50.3 & 48.7 & 48.2\end{array}\) a. For each of the given samples, calculate the mean and the standard deviation. b. Compute the coefficient of variation for each sample. Do the results surprise you? Why or why not?

An advertisement for the " 30 inch Wonder" that appeared in the September 1983 issue of the journal Packaging claimed that the 30 inch Wonder weighs cases and bags up to 110 pounds and provides accuracy to within \(0.25\) ounce. Suppose that a 50 ounce weight was repeatedly weighed on this scale and the weight readings recorded. The mean value was \(49.5\) ounces, and the standard deviation was \(0.1\). What can be said about the proportion of the time that the scale actually showed a weight that was within \(0.25\) ounce of the true value of 50 ounces? (Hint: Use Chebyshev's Rule.)

A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (in seconds) to complete the escape ("Oxygen Consumption and Ventilation During Escape from an Offshore Platform," Ergonomics [1997]: 281-292): \(\begin{array}{lllllllll}389 & 356 & 359 & 363 & 375 & 424 & 325 & 394 & 402 \\\ 373 & 373 & 370 & 364 & 366 & 364 & 325 & 339 & 393 \\ 392 & 369 & 374 & 359 & 356 & 403 & 334 & 397 & \end{array}\) a. Construct a stem-and-leaf display of the data. Will the sample mean or the sample median be larger for this data set? b. Calculate the values of the sample mean and median. c. By how much could the largest time be increased without affecting the value of the sample median? By how much could this value be decreased without affecting the sample median?

A student took two national aptitude tests. The national average and standard deviation were 475 and 100 , respectively, for the first test and 30 and 8 , respectively, for the second test. The student scored 625 on the first test and 45 on the second test. Use \(z\) scores to determine on which exam the student performed better relative to the other test takers.

The percentage of juice lost after thawing for 19 different strawberry varieties appeared in the article "Evaluation of Strawberry Cultivars with Different Degrees of Resistance to Red Scale" (Fruit Varieties Journal [1991]: \(12-17\) ): $$ \begin{array}{llllllllllll} 46 & 51 & 44 & 50 & 33 & 46 & 60 & 41 & 55 & 46 & 53 & 53 \\ 42 & 44 & 50 & 54 & 46 & 41 & 48 & & & & & \end{array} $$ a. Are there any observations that are mild outliers? Extreme outliers? b. Construct a boxplot, and comment on the important features of the plot.
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