Skip to main content
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 5.1.37a
Chapter 5, Problem 5.1.37a

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.
Data table displaying life spans of 16 tires, with values ranging from 25,314 to 48,778 miles for histogram analysis.
a. Draw a frequency histogram to display these data. Use five classes. Do the life spans appear to be normally distributed? Explain.

Verified step by step guidance
1
Step 1: Organize the data into a frequency distribution. To do this, determine the range of the data by subtracting the smallest value (25,314) from the largest value (48,778). Divide this range into five equal intervals (classes). Each interval should have the same width, calculated as (Range ÷ Number of Classes).
Step 2: Count the number of tire life spans that fall into each interval (class). This will give you the frequency for each class. For example, if the first interval is 25,314 to 30,000, count how many values fall within this range.
Step 3: Draw a frequency histogram. On the x-axis, label the intervals (classes). On the y-axis, label the frequency (number of tires in each class). Plot bars for each interval, where the height of each bar corresponds to the frequency of that interval.
Step 4: Analyze the shape of the histogram. A normal distribution typically appears as a bell-shaped curve, with most data concentrated around the mean and fewer data points in the tails. Compare the histogram to this expected shape.
Step 5: Evaluate whether the life spans appear to be normally distributed. Consider factors such as symmetry, the concentration of data around the mean (40,000 miles), and whether the histogram resembles a bell curve. Provide an explanation based on your observations.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
13m
Was this helpful?

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, indicating 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 this context, the manufacturer claims that tire life spans follow a normal distribution with a mean of 40,000 miles and a standard deviation of 4,000 miles.
Recommended video:
06:23
Using the Normal Distribution to Approximate Binomial Probabilities

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into intervals (or bins) and the frequency of data points within each interval is represented by the height of the bars. In this case, creating a histogram with five classes will help visualize the distribution of tire life spans and assess whether they align with the expected normal distribution.
Recommended video:
Guided course
05:54
Intro to Histograms

Statistical Inference

Statistical inference involves using data from a sample to make conclusions about a larger population. In this scenario, the life spans of the 16 tested tires serve as a sample to evaluate the manufacturer's claim about the population of all tires. By analyzing the histogram and applying statistical tests, one can infer whether the sample data supports the claim of normality in tire life spans.
Recommended video:
Guided course
05:53
Parameters vs. Statistics
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

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 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?