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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 5.2.19a
Chapter 5, Problem 5.2.19a

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?

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1
Identify the key parameters of the normal distribution: the mean (μ) and the standard deviation (σ). These values should be provided in Exercise 15.a. If not explicitly given, refer to the problem context to find them.
Standardize the value of 290 days using the z-score formula: z = x - μσ, where x = 290, μ is the mean, and σ is the standard deviation.
Once the z-score is calculated, use a standard normal distribution table (z-table) or statistical software to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the proportion of the population with a pregnancy length of less than 290 days.
Interpret the cumulative probability as a percentage by multiplying it by 100. This percentage represents the proportion of new mothers with a pregnancy length of less than 290 days.
Verify the result by ensuring the z-score and cumulative probability calculations are consistent with the properties of the normal distribution (e.g., probabilities should be between 0 and 1).

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Key Concepts

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

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is widely used in statistics to represent real-valued random variables whose distributions are not known. In the context of pregnancy length, it helps in understanding how pregnancy durations are distributed around the average.
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Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In this question, calculating the Z-score for a pregnancy length of 290 days will help determine how many standard deviations this length is from the average, which is essential for finding the corresponding percentile.
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Percentile

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, if a pregnancy length is at the 30th percentile, it means that 30% of pregnancies are shorter than this length. In this question, determining the percentile for a pregnancy length of less than 290 days will provide insight into how common or rare this length is among new mothers.
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

The per capita disposable income for residents of a U.S. city in a recent year is normally distributed, with a mean of about \$44,000 and a standard deviation of about \(2450. Use this information in Exercises 7–10.


Out of 800 residents, about how many would you expect to have a disposable income of between \)40,000 and \$42,000?

Textbook Question

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)

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

During a recent period of one year, the mean percent increase in value on Wednesdays of the cryptocurrency Dogecoin was 7.46%, with a standard deviation of 53.47%. Random samples of size 50 are drawn from this population and the mean of each sample is determined. (Source: Crypto Indicators)


c. What is the probability that the mean percent increase for a given sample is between −10% and 30%?

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.