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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

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.

The Normal model \(N(69,3)\) describes the distribution of male heights in the United States. 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. See page 316 for guidance. a. To be a member of the Tall Club of Silicon Valley a man must be at least 74 inches tall. What percentage of men would qualify for membership in this club? b. Suppose the Tall Club of Silicon Valley wanted to admit the tallest \(2 \%\) of men. What minimum height requirement should the club set for its membership criteria?

According to the British Medical Journal, the distribution of weights of newborn babies is approximately Normal, with a mean of 3390 grams and a standard deviation of 550 grams. Use a technology or a table to answer these questions. For each include an appropriately labeled and shaded Normal curve. a. What is the probability at newborn baby will weigh more than 4000 grams? b. What percentage of newborn babies weigh between 3000 and 4000 grams? c. A baby is classified as "low birth weight" if the baby weighs less than 2500 grams at birth. What percentage of newborns would we expect to be "low birth weight"?

Assume a standard Normal distribution. Draw a separate, well-labeled Normal curve for each part. a. Find an approximate \(z\) -score that gives a left area of \(0.7000\). b. Find an approximate \(z\) -score that gives a left area of \(0.9500\).

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?
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