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Chapter 5: Problem 13

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?

Short Answer

Expert verified
The IQ scores are: (a) approx 130.8 (b) approx 80.8 (c) approx 103.8 (d) approx lying between 70.6 and 129.4 (e) approx lying between 61.5 and 138.5

Step by step solution

01

Find the Z-Score for each percentage

We use z-table or z-score calculator to find z-scores corresponding to each percentile. (a) The z-score for the upper 2 percent is approx 2.05, as this corresponds to a cumulative probability of 0.98. (b) The z-score for the lower 10 percent is approx -1.28, as this corresponds to a cumulative probability of 0.10. (c) The z-score for the upper 60 percent is approx 0.25, as this corresponds to a cumulative probability of 0.40, since there's 40 percent on the right. (d) The z-score for the middle 95 percent will need two calculations for both sides of mean (2.5 percent on both sides). The z-score for 2.5 percent is approx -1.96, and for 97.5 percent, it's approx 1.96. (e) Similarly, the z-score for the middle 99 percent (0.5 percent on both sides) are approx -2.57 for 0.5 percent and 2.57 for 99.5 percent.
02

Convert the Z-Scores to IQ Scores

The z-score can be converted back to a value in the original dataset by multiplying it by standard deviation and adding the mean. For example, the IQ score associated with z-score of 2.05 is 2.05*15 + 100 = 130.8. Using this formula we can find the IQ scores corresponding to all the percentiles.

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Most popular questions from this chapter

Fill in the blank spaces. To identify a particular normal curve, you must know the _______(a) and _______ (b) for that distribution. To convert a particular normal curve to the standard normal curve, you must convert original scores into______ (c) scores. A z score indicates how many ______ (d) a score is ______(e) or ________(f) the mean of the distribution. Although there are infinite numbers of normal curves, there is __________ (g) standard normal curve. The standard normal curve has a _______ (h) of _____0 and a ______(i) of 1. The total area under the standard normal curve equals _________ (j) When using the standard normal table, it is important to remember that for any z score, the corresponding proportions in columns \(\mathrm{B}\) and \(\mathrm{C}\) (or columns \(\mathrm{B}^{\prime}\) and \(\mathrm{C}^{\prime}\) ) always sum to _____ (k) . Furthermore, the proportion in column B (or B') always specifies the proportion of area between the _____ (I) and the \(z\) score, while the proportion in column C (or \(C^{\prime}\) ) always specifies the proportion of area ______ (m) the z score. Although any \(z\) score can be either positive or negative, the proportions of area, specified in columns \(\mathrm{B}\) and \(\mathrm{C}\) (or columns \(\mathrm{B}^{\prime}\) and \(\mathrm{C}^{\prime}\) ), are never \(\quad\) (n) ______. Standard scores are unit-free scores expressed relative to a known ________ (0) and ______ (p) . The most important standard score is a _______(q) score. Unlike z scores, transformed standard scores usually lack ______ { (r) signs and ______ (s) points. Transformed standard scores accurately reflect the relative standing of the original _______ (t) score..

For each of the following scores, convert into transformed \(z\) scores with means and standard deviations of 50 and \(10,\) of 100 and \(15,\) and of 500 and 100 , respectively. (a) score of 34 in distribution with a mean of 41 and a standard deviation of 5 (b) score of 880 in a distribution with a mean of 700 and a standard deviation of 120 (c) score of -3 in a distribution with a mean of 12 and a standard deviation of 10

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: $$\begin{array}{|lccc|} \hline & \text { TEST SCORE } & \text { MEAN } & \text { STANDARD DEVIATION } \\\ \text { (a) } & 53 & 50 & 9 \\ \text { (b) } & 38 & 40 & 10 \\ \text { (c) } & 45 & 30 & 20 \\\\\text { (d) } & 28 & 20 & 20 \\\\\hline\end{array}$$

Scores on the Wechsler Adult Intelligence Scale (WAIS) approximate a normal curve with a mean of 100 and a standard deviation of \(15 .\) What proportion of \(1 Q\) scores are (a) above Kristen's \(125 ?\) (b) below \(82 ?\) (c) within 9 points of the mean? (d) more than 40 points from the mean?
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