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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

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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.

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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.

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Most popular questions from this chapter

Make a list of all possible outcomes for gender when a family has two children. Assume that the probability of having a boy is \(0.50\) and the probability of having a girl is also \(0.50\). Find the probability of each outcome in your list.

According to a report by the American Academy of Orthopedic Surgeons, \(29 \%\) of pedestrians admit to texting while walking. Suppose two pedestrians are randomly selected. a. If the pedestrian texts while walking, record a \(\mathrm{T}\). If not, record an \(\mathrm{N}\). List all possible sequences of Ts and Ns for the two pedestrians. b. For each sequence, find the probability that it will occur by assuming independence. c. What is the probability that neither pedestrian texts while walking? d. What is the probability that both pedestrians text while walking? e. What is the probability that exactly one of the pedestrians texts while walking?

Scores on the 2017 MCAT, an exam required for all medical school applicants, were approximately Normal with a mean score of 505 and a standard deviation of \(9.4\). a. Suppose an applicant had an MCAT score of 520 . What percentile corresponds with this score? b. Suppose to be considered at a highly selective medical school an applicant should score in the top \(10 \%\) of all test takers. What score would place an applicant in the top \(10 \%\) ?

In a standard Normal distribution, if the area to the left of a \(z\) -score is about \(0.1000\), what is the approximate \(z\) -score?

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