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Chapter 4: Problem 43

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.

Short Answer

Expert verified
The student performed better on the second test as seen by comparing the z-scores: \(z_2 = 1.875 > z_1 = 1.5\).

Step by step solution

01

Calculating the z-score for the first test

First we need to calculate the z-score for the first test. The formula for a z-score is \(z = (X - \mu) / \sigma\), where \(X\) is the raw score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.For the first test, \(X=625\), \(\mu=475\), and \(\sigma=100\). Substituting these in the formula, we get\[z_1 = (625 - 475) / 100 = 1.5\].
02

Calculating the z-score for the second test

Next, we calculate the z-score for the second test. Using the same z-score formula, now with \(X = 45\), \(\mu = 30\), and \(\sigma = 8\). Plugging these into the formula gives\[z_2= (45 - 30) / 8 = 1.875\].
03

Comparing the z-scores

We now have the z-scores for both tests, \(z_1 = 1.5\) and \(z_2 = 1.875\). The one that is higher indicates the test where the student performed better compared to the other students. In this case, \(z_2 > z_1\), so the student performed better on the second test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

National Aptitude Tests
National aptitude tests are standardized assessments used to measure a student's readiness for college and their potential for academic success. They serve as common criteria that colleges and universities can use to compare applicants from different educational backgrounds.

These tests often include a variety of subjects such as math, reading, writing, and sometimes specific areas related to a student's prospective field of study. Scores from these tests are used not only for admissions but also for scholarship decisions and placement in college courses.

The results from national aptitude tests usually follow what is known as a normal distribution, which is where concepts like mean, median, mode, and standard deviation come into play to analyze an individual's performance relative to the entire population of test-takers.
Standard Deviation
Standard deviation is a critical statistical concept and is vital when interpreting test scores from exams like national aptitude tests. It indicates the amount of variation or dispersion from the average (mean). A low standard deviation means that most numbers are close to the average, while a high standard deviation indicates that the numbers are spread out over a wider range.

In the context of test scores, a standard deviation can help us understand how diverse the abilities of the test-takers are and how significant a particular score is in comparison to the average. For example, a score far above the mean with a large standard deviation might not be as rare as the same score with a small standard deviation.
Test Performance Analysis
Test performance analysis involves evaluating how well a student has done on an exam relative to their peers. The z-score is an excellent tool for this purpose as it expresses a test score in terms of standard deviations away from the mean. It's a way of standardizing scores on different scales, allowing for direct comparison even if the tests differ in terms of difficulty or scoring.

To improve the understanding of test performance analysis, remember that a z-score of 0 means the score is exactly at the mean. Positive z-scores indicate how many standard deviations a score is above the mean, with higher values signaling better performance. Conversely, negative z-scores show how many standard deviations a score is below the mean. By comparing z-scores from multiple tests, like in our exercise, we can determine on which test the student performed relatively better, since the scale of measurement is normalized across different tests.

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Most popular questions from this chapter

The report "Who Moves? Who Stays Put? Where's Home?" (Pew Social and Demographic Trends, December 17, 2008 ) gave the accompanying data for the 50 U.S. states on the percentage of the population that was born in the state and is still living there. The data values have been arranged in order from largest to smallest. \(\begin{array}{lllllllllll}75.8 & 71.4 & 69.6 & 69.0 & 68.6 & 67.5 & 66.7 & 66.3 & 66.1 & 66.0 & 66.0 \\ 65.1 & 64.4 & 64.3 & 63.8 & 63.7 & 62.8 & 62.6 & 61.9 & 61.9 & 61.5 & 61.1\end{array}\) \(\begin{array}{lllllllllll}59.2 & 59.0 & 58.7 & 57.3 & 57.1 & 55.6 & 55.6 & 55.5 & 55.3 & 54.9 & 54.7 \\ 54.5 & 54.0 & 54.0 & 53.9 & 53.5 & 52.8 & 52.5 & 50.2 & 50.2 & 48.9 & 48.7\end{array}\) \(\begin{array}{llllll}48.6 & 47.1 & 43.4 & 40.4 & 35.7 & 28.2\end{array}\) a. Find the values of the median, the lower quartile, and the upper quartile. b. The two smallest values in the data set are 28.2 (Alaska) and 35.7 (Wyoming). Are these two states outliers? c. Construct a boxplot for this data set and comment on the interesting features of the plot.

The article "Caffeinated Energy Drinks?A Growing Problem" (Drug and Alcohol Dependence \([2009]: 1-10)\) gave the following data on caffeine concentration (mg/ounce) for eight top-selling energy drinks: a. What is the value of the mean caffeine concentration for this set of top- selling energy drinks? \(\bar{x}=9.625\) b. Coca-Cola has \(2.9 \mathrm{mg} /\) ounce of caffeine and Pepsi Cola has \(3.2 \mathrm{mg} /\) ounce of caffeine. Write a sentence explaining how the caffeine concentration of topselling energy drinks compares to that of these colas.

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 \(\begin{array}{llllllllll}\text { S } & \text { S } & \text { F } & \text { S } & \text { S } & \text { S } & \text { F } & \text { F } & \text { S } & \text { S }\end{array}\) a. What is the value of the sample proportion of successes? b. Replace each \(S\) with a 1 and each \(F\) with a 0 . Then calculate \(\bar{x}\) for this numerically coded sample. How does \(\bar{x}\) compare to \(\hat{p}\) ? 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 \(\hat{p}=.80\) for the entire sample of 25 patients?

The Insurance Institute for Highway Safety (www.iihs.org, June 11,2009 ) published data on repair costs for cars involved in different types of accidents. In one study, seven different 2009 models of mini- and micro-cars were driven at 6 mph straight into a fixed barrier. The following table gives the cost of repairing damage to the bumper for each of the seven models: $$ \begin{array}{lc} \text { Model } & \text { Repair Cost } \\ \hline \text { Smart Fortwo } & \$ 1,480 \\ \text { Chevrolet Aveo } & \$ 1,071 \\ \text { Mini Cooper } & \$ 2,291 \\ \text { Toyota Yaris } & \$ 1,688 \\ \text { Honda Fit } & \$ 1,124 \\ \text { Hyundai Accent } & \$ 3,476 \\ \text { Kia Rio } & \$ 3,701 \\ \hline \end{array} $$ a. Compute the values of the variance and standard deviation. The standard deviation is fairly large. What does this tell you about the repair costs? b. The Insurance Institute for Highway Safety (referenced in the previous exercise) also gave bumper repair costs in a study of six models of minivans (December 30,2007 ). Write a few sentences describing how mini- and micro-cars and minivans differ with respect to typical bumper repair cost and bumper repair cost variability. $$ \begin{array}{lr} \text { Model } & \text { Repair Cost } \\ \hline \text { Honda Odyssey } & \$ 1,538 \\ \text { Dodge Grand Caravan } & \$ 1,347 \\ \text { Toyota Sienna } & \$ 840 \\ \text { Chevrolet Uplander } & \$ 1,631 \\ \text { Kia Sedona } & \$ 1,176 \\ \text { Nissan Quest } & \$ 1,603 \\ \hline \end{array} $$

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. \(610 \mathrm{wpm}\)
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