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Chapter 6: Problem 33

SAT math scores follow a normal distribution with an approximate \(\mu=500\) and \(\sigma=100\). Also ACT math scores follow a normal distrubution with an approximate \(\mu=21\) and \(\sigma=4.7\). You are an admissions officer at a university and have room to admit one more student for the upcoming year. Joe scored 600 on the SAT math exam, and Kate scored 25 on the \(\mathrm{ACT}\) math exam. If you were going to base your decision solely on their performances on the exams, which student should you admit? Explain.

Short Answer

Expert verified
Admit Joe, as his SAT z-score (1.00) is higher than Kate's ACT z-score (0.851).

Step by step solution

01

Understanding the Problem

We need to determine who performed better relative to their respective test populations. This will be done by calculating the z-scores for both Joe's SAT score and Kate's ACT score.
02

Calculate Joe's SAT Z-Score

First, calculate Joe's z-score using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x = 600 \), \( \mu = 500 \), and \( \sigma = 100 \). \[ z_{SAT} = \frac{600 - 500}{100} = \frac{100}{100} = 1 \]
03

Calculate Kate's ACT Z-Score

Next, calculate Kate's z-score using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x = 25 \), \( \mu = 21 \), and \( \sigma = 4.7 \).\[ z_{ACT} = \frac{25 - 21}{4.7} = \frac{4}{4.7} \approx 0.851 \]
04

Compare Z-Scores

The next step is to compare the z-scores. Joe's z-score is 1.00, and Kate's z-score is approximately 0.851. A higher z-score indicates a better performance relative to the population.
05

Decision Based on Z-Scores

Since Joe's z-score (1.00) is greater than Kate's z-score (0.851), Joe performed better relative to the SAT population compared to Kate's performance relative to the ACT population.

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Key Concepts

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

Z-Score Calculation
The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. By converting scores into z-scores, we can determine how far and in which direction a score deviates from the mean. This calculation is vital when comparing different data sets, like SAT and ACT scores.
  • Formula: \( z = \frac{x - \mu}{\sigma} \)
  • \( x \) is an individual data point (e.g., an SAT or ACT score)
  • \( \mu \) is the mean of the data set
  • \( \sigma \) is the standard deviation
A higher absolute z-score indicates the score is farther from the mean. Positive z-scores mean the value is above the mean, while negative indicates it's below. Calculating z-scores allows us to compare scores from different distributions effectively.
Normal Distribution
The normal distribution, often referred to as the "bell curve," is a key concept in statistics. It's a way to describe data that clusters around a mean.
  • It is symmetrical about the mean.
  • Most data falls within three standard deviations from the mean.
  • The mean, median, and mode are all equal.
This distribution is incredibly important as it allows statisticians to make predictions about data. For tests like the SAT and ACT, which follow a normal distribution, we can calculate the z-score to understand relative performance. Understanding this distribution helps in identifying how typical or atypical a score is when compared to the rest of the population.
SAT and ACT Scores
SAT and ACT exams are standardized tests used widely in college admissions in the United States. Each follows a normal distribution, meaning they鈥檙e designed to compare participants based on relative performance.
  • The SAT math section has a mean \( \mu = 500 \) and a standard deviation \( \sigma = 100 \).
  • The ACT math section has a mean \( \mu = 21 \) and a standard deviation \( \sigma = 4.7 \).
By converting a score into a z-score, we can see how a specific individual's performance compares not just within the test itself, but across different types of assessments. This comparative performance analysis is crucial when making admissions decisions based solely on test results.
Educational Assessment
Educational assessment involves evaluating and analyzing students' performance to guide decision-making processes in education. It uses tools such as standardized tests to measure various competencies.
  • Standardized tests like the SAT and ACT allow for uniform metrics of evaluation.
  • They enable institutions to maintain consistent standards for admissions.
  • Z-scores and normal distributions help in comparing different test scores fairly.
These assessments provide a way to benchmark student abilities and readiness for further education. By understanding z-scores and normal distribution, educators can make informed decisions on student admissions based on standardized test performances.

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Most popular questions from this chapter

In the National Basketball Association, the top free throw shooters usually have probability of about 0.90 of making any given free throw. a. During a game, one such player (Dirk Nowitzki) shot 10 free throws. Let \(X=\) number of free throws made. What must you assume for \(X\) to have a binomial distribution? (Studies have shown that such assumptions are well satisfied for this sport.) b. Specify the values of \(n\) and \(p\) for the binomial distribution of \(X\) in part a. c. Find the probability that he made (i) all 10 free throws (ii) 9 free throws and (iii) more than 7 free throws.

Movies sample Five of the 20 movies running in movie theatres this week are comedies. A selection of four movies are picked at random. Does \(X=\) the number of movies in the sample which are comedies have the binomial distribution with \(n=4\) and \(p=0.25 ?\) Explain why or why not.

A professor of statistics wants to prepare a test paper by selecting five questions randomly from an online test bank available for his course. In the test bank, the proportion of questions labeled "HARD" is 0.3 . a. Find the probability that all the questions selected for the test are labeled HARD. b. Find the probability that none of the questions selected for the test is labeled HARD. c. Find the probability that less than half of the questions selected for the test are labeled HARD.

For a normal distribution, a. Find the \(z\) -score for which a total probability of 0.04 falls more than \(z\) standard deviations (in either direction) from the mean, that is, below \(\mu-z \sigma\) or above \(\mu+z \sigma\) b. For this \(z\) -score, explain why the probability of values more than \(z\) standard deviations below the mean is 0.02 . c. Explain why \(\mu+z \sigma\) is the 2 nd percentile.

Ideal number of children Let \(X\) denote the response of a randomly selected person to the question, "What is the ideal number of children for a family to have?" The probability distribution of \(X\) in the United States is approximately as shown in the table, according to the gender of the person asked the question. $$ \begin{array}{lcc} \hline \begin{array}{l} \text { Probability Distribution of } \boldsymbol{X}=\text { Ideal Number } \\\ \text { of Children } \end{array} \\ \hline \boldsymbol{x} & \mathbf{P}(\boldsymbol{x}) \text { Females } & \mathbf{P}(\boldsymbol{x}) \text { Males } \\ \hline 0 & 0.01 & 0.02 \\ 1 & 0.03 & 0.03 \\ 2 & 0.55 & 0.60 \\ 3 & 0.31 & 0.28 \\ 4 & 0.11 & 0.08 \\ \hline \end{array} $$ a. Show that the means are similar, 2.50 for females and 2.39 for males. b. The standard deviation for the females is 0.770 and 0.758 for the males. Explain why a practical implication of the values for the standard deviations is that males hold slightly more consistent views than females about the ideal family size.
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