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Chapter 5: Problem 19

When describing test results, someone objects to the conversion of raw scores into standard scores, claiming that this constitutes an arbitrary change in the value of the test score. How might you respond to this objection?

Short Answer

Expert verified
The conversion of raw scores into standard scores is not an arbitrary process but a standardization, which allows scores to be compared on a common scale. These scores maintain the initial raw score's relative position within the distribution, enabling a meaningful comparison across different scales of measure.

Step by step solution

01

Understanding the Concept of Standard Scores

Standard scores are statistical measures that find extensive use in comparing data from different scales. They provide a measure of how far an individual raw score is from the mean, expressed in terms of standard deviations. This conversion allows for the measurement of scores on a common scale, hence eliminating any inequalities among different scales of measurement, and thereby rationalizing the comparison of scores.
02

Addressing the Objection

While it may seem that the conversion of raw scores to standard scores could be arbitrary, it is indeed not. The conversion process ensures that each raw score is adjusted in reference to the entire data set's distribution. Thus, the standard score retains the original raw score's relative position within its data set. So, rather than an arbitrary change, it is a standardization allowing for representative measurement on a common scale.
03

Providing an Illustration

Consider two students, one scoring 85 out of 100 and the other 170 out of 200 in two different tests. Although the raw scores seem different, both have achieved 85% in their respective tests. Converting these raw scores into standard scores would reflect this fact and allow for a holistic comparison.

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Most popular questions from this chapter

IQ scores on the WAIS test approximate a normal curve with a mean of 100 and a standard deviation of \(15 .\) What IQ score is identified with (a) the upper 2 percent, that is, 2 percent to the right (and 98 percent to the left)? (b) the lower 10 percent? (c) the upper 60 percent? (d) the middle 95 percent? [Remember, the middle 95 percent straddles the line perpendicular to the mean (or the 50th percentile), with half of 95 percent, or 47.5 percent, above this line and the remaining 47.5 percent below this line.] (e) the middle 99 percent?

Assume that the burning times of electric light bulbs approximate a normal curve with a mean of 1200 hours and a standard deviation of 120 hours. If a large number of new lights are installed at the same time (possibly along a newly opened freeway), at what time will (a) 1 percent fail? (Hint: This splits the total area into .0100 to the left and .9900 to the right.) (b) 50 percent fail? (c) 95 percent fail? (d) If a new inspection procedure eliminates the weakest 8 percent of all lights before they are marketed, the manufacturer can safely offer customers a money-back guarantee on all lights that fail before ______ hours of burning time.

Suppose that the burning times of electric light bulbs approximate a normal curve with a mean of 1200 hours and a standard deviation of 120 hours. What proportion of lights burn for (a) less than 960 hours? (b) more than 1500 hours? (c) within 50 hours of the mean? (d) between 1300 and 1400 hours?

Express each of the following scores as a z score: (a) Margaret's \(1 Q\) of \(135,\) given a mean of 100 and a standard deviation of 15 (b) a score of 470 on the SAT math test, given a mean of 500 and a standard deviation of 100 (c) a daily production of 2100 loaves of bread by a bakery, given a mean of 2180 and a standard deviation of 50 (d) Sam's height of 69 inches, given a mean of 69 and a standard deviation of 3 (e) a thermometer-reading error of -3 degrees, given a mean of 0 degrees and a standard deviation of 2 degrees

Convert each of the following test scores to \(z\) scores: $$\begin{array}{|lccc|} \hline & \text { TEST SCORE } & \text { MEAN } & \text { STANDARD DEVIATION } \\\ \text { (a) } & 53 & 50 & 9 \\ \text { (b) } & 38 & 40 & 10 \\ \text { (c) } & 45 & 30 & 20 \\\\\text { (d) } & 28 & 20 & 20 \\\\\hline\end{array}$$
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