/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 The Normal model \(N(150,10)\) d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 42

The Normal model \(N(150,10)\) describes the distribution of scores on the LSAT, a standardized test required by most law schools. Which of the following questions asks for a probability, and which asks for a measurement? Identify the type of problem and then answer the given question. a. A law school applicant scored at the 60 th percentile on the LSAT. What was the applicant's LSAT score? b. A law school applicant scored 164 on the LSAT. This applicant scored higher than what percentage of LSAT test takers?

Short Answer

Expert verified
For subpart (a) the law school applicant in the 60th percentile scored 152.5 on the LSAT. In subpart (b) an applicant who scored 164 scored higher than approximately 92% of other LSAT test takers.

Step by step solution

01

Calculation of LSAT Score for 60th Percentile

To calculate the score at the 60th percentile, it's necessary to convert the percentile value in terms of standard deviations from the mean using z-score tables. The value for the z-score of 60th percentile is approximately 0.25. We then use the formula for a z-score: z = (X - 碌)/蟽 where X is the score we want to find, 碌 is the mean, 蟽 is the standard deviation and z is the z-score. Solving for X we get X = z*蟽 + 碌. Substituting z = 0.25, 蟽 = 10, and 碌 = 150 we get X = 0.25*10 + 150 which gives us X = 152.5. So, a student scoring at the 60th percentile scored 152.5 on LSAT.
02

Calculation of Percentile for LSAT Score of 164

To calculate the percentile for the score of 164, we need to find the z-score using the formula: z = (X - 碌)/蟽. Substituting X = 164, 碌 = 150 and 蟽 = 10, we find a z-score of 1.4. Referring to the z-score tables, a z-score of 1.4 corresponds to approximately the 92nd percentile. So, a score of 164 is higher than about 92% of LSAT test takers.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Z-scores
The concept of a z-score is essential in statistics for understanding where a particular data point sits relative to the mean of a data set. A z-score indicates how many standard deviations a point is from the mean.

Here's how a z-score works:
  • If a z-score is 0, the data point is exactly at the mean.
  • Positive z-scores indicate values above the mean.
  • Negative z-scores show points below the mean.
To calculate a z-score, use the formula: \[ z = \frac{X - \mu}{\sigma} \]where:
  • \(X\) is the score you're investigating,
  • \(\mu\) is the mean of the data, and
  • \(\sigma\) is the standard deviation.
In our LSAT example, finding the z-score was crucial in determining percentiles and specific scores.
Deciphering Percentiles
Percentiles are helpful statistical tools that indicate the position of a score within a data set. They tell us what percentage of observations fall below a certain point.

For example:
  • The 60th percentile means a score is higher than 60% of the data.
  • A score in the 92nd percentile means it's higher than 92% of the data points.
In our exercise, we used percentiles to determine the positions of scores on a standardized test, the LSAT. By using a z-score chart, one can easily find the percentile associated with a specific z-score.
The Role of Standard Deviation
Standard deviation (\(\sigma\)) is a measure that quantifies the amount of variation in a set of data values. In simpler terms, it shows how spread out the numbers in a data set are from the mean.

This measurement is crucial because:
  • A small standard deviation means the data points are close to the mean.
  • A large standard deviation means the data are spread out over a wider range of values.
In our LSAT scenario, the standard deviation of 10 helps us understand how much an individual score can vary from the average score of 150.
Unpacking the Mean
The mean of a data set is its average. It is calculated by summing all data points and then dividing by the number of points. In the LSAT example, the mean score is 150.

The mean is important because:
  • It provides a central value for the dataset.
  • It is used as a reference point for calculating the z-score.
However, the mean alone doesn't provide full picture of the dataset, as it doesn't account for the variability or spread of the scores.
Probability Measurement in Context
Probability measurement in statistics can often mean different things, depending on context. In the case of normal distributions, it often involves the area under the curve, signifying the likelihood of a specific outcome.

Probability can refer to:
  • The chance of scoring in a specific range.
  • The likelihood that a certain percentile rank corresponds to a given score.
When answering questions like in our LSAT example, measuring probability involves translating scores into z-scores and then looking up corresponding percentiles or scores using z-score tables.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Assume that college women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. On the horizontal axis of the graph, indicate the heights that correspond to the \(z\) -scores provided. (See the labeling in Exercise 6.14.) Use only the Empirical Rule to choose your answers. Sixty inches is 5 feet, and 72 inches is 6 feet. a. Roughly what percentage of women's heights are greater than \(72.5\) inches? i. almost all iii. \(50 \%\) v. about \(0 \%\) ii. \(75 \%\) iv. \(25 \%\) b. Roughly what percentage of women's heights are between 60 and 70 inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) c. Roughly what percentage of women's heights are between 65 and \(67.5\) inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) d. Roughly what percentage of women's heights are between \(62.5\) and \(67.5\) inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) e. Roughly what percentage of women's heights are less than \(57.5\) inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) f. Roughly what percentage of women's heights are between 65 and 70 inches? i. almost all iii. \(47.5 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\)

Length of Pregnancy Assume that the lengths of pregnancy for humans is approximately Normally distributed, with a mean of 267 days and a standard deviation of 10 days. Use the Empirical Rule to answer the following questions. Do not use the technology or the Normal table. Begin by labeling the horizontal axis of the graph with lengths, using the given mean and standard deviation. Three of the entries are done for you. a. Roughly what percentage of pregnancies last more than 267 days? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) b. Roughly what percentage of pregnancies last between 267 and 277 days? i. \(34 \%\) iii. \(2.5 \%\) ii. \(17 \%\) iv. \(50 \%\) c. Roughly what percentage of pregnancies last less than 237 days? i. almost all iii. \(34 \%\) ii. \(50 \%\) iv. about \(0 \%\) d. Roughly what percentage of pregnancies last between 247 and 287 days? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) e. Roughly what percentage of pregnancies last longer than 287 days? i. \(34 \%\) iii. \(2.5 \%\) ii. \(17 \%\) iv. \(50 \%\) f. Roughly what percentage of pregnancies last longer than 297 days? i. almost all iii. \(34 \%\) ii. \(50 \%\) iv. about \(0 \%\)

Quantitative SAT scores are approximately Normally distributed with a mean of 500 and a standard deviation of 100 . On the horizontal axis of the graph, indicate the SAT scores that correspond with the provided \(z\) -scores. (See the labeling in Exercise 6.14.) Answer the questions using only your knowledge of the Empirical Rule and symmetry. a. Roughly what percentage of students earn quantitative SAT scores greater than \(500 ?\) i. almost all iii. \(50 \%\) v. about \(0 \%\) ii. \(75 \%\) iv. \(25 \%\) b. Roughly what percentage of students earn quantitative SAT scores between 400 and 600 ? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) c. Roughly what percentage of students earn quantitative SAT scores greater than 800 ? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) d. Roughly what percentage of students earn quantitative SAT scores less than \(200 ?\) i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) e. Roughly what percentage of students earn quantitative SAT scores between 300 and 700 ? i. almost all iii. \(68 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\) f. Roughly what percentage of students earn quantitative SAT scores between 700 and 800 ? i. almost all iii. \(68 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\)

Assume college women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. Choose the Stat- \(-\) Crunch output for finding the percentage of college women who are taller than 67 inches and report the correct percentage. Round to one decimal place. a. b.

According to the Centers of Disease Control and Prevention, \(44 \%\) of U.S. households still had landline phone service. Suppose a random sample of 60 U.S. households is taken. a. Find the probability that exactly 25 of the households sampled still have a landline. b. Find the probability that more than 25 households still have a landline. c. Find the probability that at least 25 households still have a landline. d. Find the probability that between 20 and 25 households still have a landline.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.