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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.R.41
Chapter 5, Problem 5.R.41

In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.


P85

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1
Identify the given information: The problem asks for the z-score corresponding to the 85th percentile (P85). This means that 85% of the data lies below this z-score in a standard normal distribution.
Understand the relationship between percentiles and cumulative probabilities: The 85th percentile corresponds to a cumulative probability of 0.85 in the standard normal distribution.
Use the Standard Normal Table or technology: Locate the cumulative probability of 0.85 in the Standard Normal Table (or use statistical software or a calculator with a z-score function). The table or tool will provide the z-score corresponding to this cumulative probability.
Interpret the z-score: The z-score represents the number of standard deviations the value is above or below the mean. Since the cumulative probability is greater than 0.5, the z-score will be positive, indicating it is above the mean.
Verify the result: Double-check the cumulative probability in the table or software to ensure the z-score corresponds to 0.85. This step ensures accuracy in your solution.

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Key Concepts

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

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. Z-scores indicate how many standard deviations an element is from the mean, allowing for comparison across different datasets.
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Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve and is used to determine probabilities and percentiles for normally distributed data. The area under the curve corresponds to probabilities, making it essential for z-score calculations.
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Cumulative Area

Cumulative area refers to the total area under the curve of a probability distribution up to a certain z-score. It represents the probability that a random variable drawn from the distribution will be less than or equal to that z-score. This concept is crucial for finding percentiles and understanding the distribution of data in statistics.
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Related Practice
Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (c) greater than 5900 MMT CO2 eq. Compare your answers with those in Exercise 33.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (a) between 5.1 and 15.7 picomoles per liter. Compare your answers with those in Exercise 34.

Textbook Question

What braking distance represents the first quartile?

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean MCAT total score in a recent year is 500.9. A random sample of 32 MCAT total scores is selected. What is the probability that the mean score for the sample is (a) less than 503? Assume sigma=10.6.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (b) between 6000 and 6500 MMT CO2 eq. Compare your answers with those in Exercise 33.

Textbook Question

In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.


P(x < 60)