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Chapter 6: Problem 50

The distribution of grade point averages GPAs for medical school applicants in 2017 were approximately Normal, with a mean of \(3.56\) and a standard deviation of 0.34. Suppose a medical school will only consider candidates with GPAs in the top \(15 \%\) of the applicant pool. An applicant has a GPA of \(3.71\). Does this GPA fall in the top \(15 \%\) of the applicant pool?

Short Answer

Expert verified
No, a GPA of 3.71 does not fall in the top 15% of the applicant pool.

Step by step solution

01

Define the given data

Given that the mean (\(\mu\)) is 3.56 and the standard deviation (\(\sigma\)) is 0.34. The GPA to test is 3.71. For the given normal distribution, it needs to be determined if 3.71 falls into the top 15%
02

Calculate the z-score for the given GPA

The z-score is calculated using the formula \(Z = \frac{X - \mu}{\sigma}\) where X is the score we're interested in. So, plug in the values: \(Z = \frac{3.71 - 3.56}{0.34} = 0.44\)
03

Conversion of z-score to percentile

A z-score of 0.44 corresponds approximately to the 67th percentile (This can be obtained from a standard Z-table or using statistical software).
04

Determine if the GPA is in the top 15%

To be in the top 15%, a GPA should be in the 85th percentile or higher, as 100 - 15 = 85. As the GPA we're testing corresponds to the 67th percentile, it is not in the top 15%.

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Key Concepts

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

z-score calculation
Understanding how to calculate the z-score is crucial when dealing with the normal distribution. The z-score tells us how many standard deviations a data point is from the mean. It is calculated with the formula:
\[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the value of the data point, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.
- **Example:** In our context, let's calculate for a GPA of 3.71 with a mean GPA of 3.56 and a standard deviation of 0.34, the z-score would be \( Z = \frac{3.71 - 3.56}{0.34} \).
- This results in a z-score of 0.44, meaning the GPA is 0.44 standard deviations above the mean.
This z-score helps us understand where this GPA falls in relation to the entire distribution of GPAs.
percentile
A percentile indicates the position or rank of a score in a distribution. If a score is at the 85th percentile, it means it is higher than 85% of the other scores. To find the percentile, one can use a z-table or statistical software.
- **Example of Use:** When we calculated a z-score of 0.44 for the GPA, we needed to convert this into a percentile.
- Using a z-table, the z-score of 0.44 converts approximately to the 67th percentile.
This tells us that the GPA of 3.71 is higher than approximately 67% of all GPAs in the applicant pool.
Understanding percentiles helps in making informed decisions, especially when a cut-off point is needed as seen in the evaluation of GPA requirements for medical school applications.
standard deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In a normal distribution:
- **The Role of Standard Deviation:** A small standard deviation means the data points are close to the mean. A large standard deviation indicates that the data are spread out over a wider range of values.
In our problem, the standard deviation is 0.34, which reflects how much students' GPAs vary around the mean GPA of 3.56.
Understanding standard deviation is fundamental since it impacts our calculation of the z-score, which in turn helps us determine the percentile rank of a specific GPA.

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Most popular questions from this chapter

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 to support marijuana legalization, give or take

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Standard Normal Use technology or a Normal table to find each of the following. Include an appropriately labeled sketch of the Normal curve for each part with the appropriate area shaded. a. Find the probability that a \(z\) -score will be \(2.03\) or less. b. Find the probability that a \(z\) -score will be \(-1.75\) or more. c. Find the probability that a \(z\) -score will be between \(-1.25\) and \(1.40\).

According to a Pew poll, \(67 \%\) of Americans believe that jury duty is part of good citizenship. Suppose 500 Americans are randomly selected. a. Find the probability that more than half believe that jury duty is part of good citizenship. b. In a group of 500 Americans, how many would we expect hold this belief? c. Would it be surprising to find that more than 450 out of the 500 American randomly selected held this belief? Why or why not?
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