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Chapter 6: Problem 49

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 \(10 \%\) ?

Short Answer

Expert verified
To determine the percentile of a test score, calculate a Z score and use a standard normal distribution table to match the Z score to a percentile. To find the score that corresponds to a given percentile, determine the Z score for that percentile and then calculate the corresponding test score using the given mean and standard deviation.

Step by step solution

01

Understanding Each Part of the Problem

The problem consists of two parts. In the first part, it is required to determine the percentile of a 520 MCAT score. In the second one, the MCAT score required to be in the top 10% test takers is to be found. The mean score is provided as 505 and the standard deviation as 9.4 from the 2017 MCAT.
02

Calculating the percentile for a 520 MCAT score

The formula for calculating the Z score, Z = (X - 渭) / 蟽, will be used. Here, X is the value of the score (520), 渭 is the mean (505), and 蟽 is the standard deviation (9.4). Substitute these values into the formula and calculate the Z score. Once the Z score is calculated, look up this value in a standard normal distribution table, or use a calculator or computer software to find the percentile. A standard normal distribution table would provide the percentage of individuals who scored below this Z score (520 MCAT score). This will be the percentile for the 520 MCAT score.
03

Calculating the MCAT score for the top 10 percent

To calculate the MCAT score that an applicant would need to be in the top 10 percent, calculate the Z score that corresponds to a percentile of 90% (or 0.9 in decimal form) from a standard normal table. Using the Z = (X - 渭) / 蟽 formula, rearrange and solve for X. The X value obtained will be the MCAT score an applicant would need to be in the top 10 percent.
04

Interpret the Results

After each calculation, the results must clearly interpret to answer the problem. The percentile derived from the first calculation explains the standing of a test taker with a 520 score in reference to all test takers. The MCAT score from the second calculation states the score an applicant must have to be among the top 10 percent of all test takers.

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Most popular questions from this chapter

Babies in the United States have a mean birth length of \(20.5\) inches with a standard deviation of \(0.90\) inch. The shape of the distribution of birth lengths is approximately Normal. a. How long is a baby born at the 20 th percentile? b. How long is a baby born at the 50 th percentile? c. How does your answer to part b compare to the mean birth length? Why should you have expected this?

Toronto drivers have been going to small towns in Ontario (Canada) to take the drivers' road test, rather than taking the test in Toronto, because the pass rate in the small towns is \(90 \%\), which is much higher than the pass rate in Toronto. Suppose that every day, 100 people independently take the test in one of these small towns. a. What is the number of people who are expected to pass? b. What is the standard deviation for the number expected to pass? c. After a great many days, according to the Empirical Rule, on about 95\% of these days the number of people passing the test will be as low as _______ and as high as _______. d. If you found that on one day, 89 out of 100 passed the test, would you consider this to be a very high number?

Use the table or technology to find the answer to each question. Include an appropriately labeled sketch of the Normal curve for each part. Shade the appropriate region. A section of the Normal table is provided. a. Find the area in a Standard Normal curve to the left of \(1.13 .\) b. Find the area in a Standard Normal curve to the right of \(1.13 .\)

Weather New York City's mean minimum daily temperature in February is \(27^{\circ} \mathrm{F}\) (http://www.ny.com). Suppose the standard deviation of the minimum temperature is \(6^{\circ} \mathrm{F}\) and the distribution of minimum temperatures in February is approximately Normal. What percentage of days in February has minimum temperatures below freezing \(\left(32^{\circ} \mathrm{F}\right)\) ?

Support for the legalization of marijuana has continued to grow among Americans. A 2017 Gallup poll found that \(64 \%\) of Americans now say that marijuana use should be legal. Suppose a random sample of 150 Americans is selected. a. Find the probability that at most 110 people support marijuana legalization. b. Find the probability that between 90 and 110 support marijuana legalization. c. Complete this sentence: In a group of 150 , we would expect _____ support marijuana legalization, give or take ____.
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