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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
Chapter 5, Problem 5.1.11

Graphical Analysis In Exercises 11–16, determine whether the graph could represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal distribution, estimate the mean and standard deviation.
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Step 1: Observe the graph and determine its shape. A normal distribution is characterized by a symmetric bell-shaped curve, where the highest point (the peak) corresponds to the mean, and the curve tapers off equally on both sides.
Step 2: Check for symmetry. Analyze whether the graph is symmetric around a central value. In this case, the graph appears to be symmetric around x = 50, which suggests it could represent a normal distribution.
Step 3: Estimate the mean. The mean of a normal distribution is the value at the center of the graph, where the peak occurs. Based on the graph, the mean is approximately x = 50.
Step 4: Estimate the standard deviation. The standard deviation can be estimated by observing the spread of the graph. For a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Identify the points where the curve starts to taper off significantly, which appear to be around x = 48 and x = 52. The standard deviation is roughly half the distance between these points and the mean.
Step 5: Conclude whether the graph represents a normal distribution. Based on the symmetry and bell-shaped curve, the graph appears to represent a variable with a normal distribution. The mean is approximately 50, and the standard deviation can be estimated using the spread of the data.

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Key Concepts

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

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (average) and the standard deviation (spread). In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, and about 95% falls within two standard deviations.
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Mean and Standard Deviation

The mean is the average value of a dataset, calculated by summing all values and dividing by the number of observations. The standard deviation measures the dispersion of data points around the mean, indicating how spread out the values are. In a normal distribution, the mean, median, and mode are all equal, and the standard deviation helps determine the width of the bell curve.
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Graphical Analysis

Graphical analysis involves interpreting visual representations of data, such as histograms or density plots, to identify patterns, trends, and distributions. In the context of normal distribution, one looks for symmetry, a single peak (unimodal), and the characteristic bell shape. Analyzing the graph can help estimate the mean and standard deviation based on the shape and spread of the data.
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Related Practice
Textbook Question

In Exercises 39 and 40, determine whether the finite correction factor should be used. If so, use it in your calculations when you find the probability.


Parking Infractions In a sample of 1000 fines issued by the City of Toronto for parking infractions in September of 2020, the mean fine was \$49.83 and the standard deviation was \$52.15. A random sample of size 60 is selected from this population. What is the probability that the mean fine is less than \$40?

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Textbook Question

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

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Textbook Question

Testing a Drug A drug manufacturer claims that a drug cures a rare skin disease 75% of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted. Use this information in Exercises 31 and 32.


Find the probability that the claim will be accepted, assuming that the actual probability that the drug cures the skin disease is 65%.

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Textbook Question

In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.


A gear has been designed to have a diameter of 3 inches. The standard deviation of the process is 0.2 inch.


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Textbook Question

Finding Probabilities for Sampling Distributions In Exercises 29–32, find the indicated probability and interpret the results.


Dow Jones Industrial Average From 1975 through 2020, the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 32 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 700? Assume sigma=1540

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Textbook Question

True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


A sampling distribution is normal only when the population is normal.

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