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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.2.9a
Chapter 5, Problem 5.2.9a

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.


MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (a) less than 490. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

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Step 1: Understand the problem. The MCAT scores are normally distributed with a mean (μ) of 500.9 and a standard deviation (σ) of 10.6. We are tasked with finding the probability that a randomly selected student has a score less than 490. This involves calculating the cumulative probability for a normal distribution.
Step 2: Standardize the score. To find the probability, we first convert the raw score (490) into a z-score using the formula: z = (X - μ) / σ. Here, X is the raw score, μ is the mean, and σ is the standard deviation. Substitute the given values into the formula.
Step 3: Use the z-score to find the cumulative probability. Once the z-score is calculated, use a z-table, statistical software, or a calculator to find the cumulative probability corresponding to the z-score. This cumulative probability represents the probability that a score is less than 490.
Step 4: Interpret the result. Compare the calculated probability to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05 (5%).
Step 5: Repeat for other parts if needed. If there are additional parts to the problem (e.g., parts (b) and (c)), follow the same process: standardize the score, find the cumulative probability, and interpret the result. Clearly explain why any event is considered unusual based on the probability threshold.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In the context of the MCAT scores, the normal distribution allows us to understand how scores are spread around the average score of 500.9.
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Z-Scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. For the MCAT scores, calculating the Z-score for a score of 490 will help determine how unusual this score is compared to the average, allowing us to find the corresponding probability.
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Probability Calculation

Probability calculation in statistics involves determining the likelihood of a particular outcome occurring within a defined set of data. For normally distributed data, this often involves using Z-scores and standard normal distribution tables or technology to find the area under the curve. In this case, we will calculate the probability of a medical student scoring less than 490 using the mean and standard deviation provided.
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Related Practice
Textbook Question

A certain stock's daily percent return on Fridays has a mean of 3.12% and a standard deviation of 41.25%. If random samples of 40 days are selected and the mean return for each sample is calculated, what is the probability that a sample mean is greater than 17%?

Textbook Question

[APPLET] Milk Consumption You are performing a study about weekly per capita milk consumption. A previous study found weekly per capita milk consumption to be normally distributed, with a mean of 48.7 fluid ounces and a standard deviation of 8.6 fluid ounces. You randomly sample 30 people and record the weekly milk consumptions shown below.

a. Draw a frequency histogram to display these data. Use seven classes. Do the consumptions appear to be normally distributed? Explain.

Textbook Question

Ice Cream The weights of ice cream cartons are normally distributed with a mean weight of 10 ounces and a standard deviation of 0.5 ounce.


a. What is the probability that a randomly selected carton has a weight greater than 10.21 ounces?


Textbook Question

Pregnancy Length Use the normal distribution in Exercise 15.


a. What percent of the new mothers had a pregnancy length of less than 290 days?

Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.


Red Blood Cell Count The red blood cell counts (in millions of cells per microliter) for a population of adult males can be approximated by a normal distribution, with a mean of 5.4 million cells per microliter and a standard deviation of 0.4 million cells per microliter.


a. What is the minimum red blood cell count that can be in the top 25% of counts?


Textbook Question

Manufacturer Claims You work for a consumer watchdog publication and are testing the advertising claims of a tire manufacturer. The manufacturer claims that the life spans of the tires are normally distributed, with a mean of 40,000 miles and a standard deviation of 4000 miles. You test 16 tires and record the life spans shown below.

a. Draw a frequency histogram to display these data. Use five classes. Do the life spans appear to be normally distributed? Explain.