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 5.4.38a
Chapter 5, Problem 5.4.38a

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?

Verified step by step guidance
1
Step 1: Identify the key parameters of the normal distribution. The mean (μ) is 10 ounces, and the standard deviation (σ) is 0.5 ounces. The problem asks for the probability that a randomly selected carton has a weight greater than 10.21 ounces.
Step 2: Standardize the value 10.21 ounces to a z-score using the formula: z=x-μσ, where x is the value of interest (10.21), μ is the mean (10), and σ is the standard deviation (0.5).
Step 3: Substitute the values into the z-score formula: z=10.21-100.5. Simplify the calculation to find the z-score.
Step 4: Use a standard normal distribution table or a statistical software to find the cumulative probability corresponding to the calculated z-score. This gives the probability that a randomly selected carton has a weight less than 10.21 ounces.
Step 5: Subtract the cumulative probability from 1 to find the probability that a randomly selected carton has a weight greater than 10.21 ounces, as the problem specifically asks for the probability of being greater than this value.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
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, depicting 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 weights of ice cream cartons follow a normal distribution with a mean of 10 ounces and a standard deviation of 0.5 ounces.
Recommended video:
06:23
Using the Normal Distribution to Approximate Binomial Probabilities

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 scenario, calculating the Z-score for a weight of 10.21 ounces will help determine how many standard deviations away this weight is from the mean, which is essential for finding the corresponding probability.
Recommended video:
Guided course
06:31
Z-Scores From Given Probability - TI-84 (CE) Calculator

Probability Calculation

Probability calculation in statistics involves determining the likelihood of a specific event occurring. For normally distributed data, this often requires using Z-scores and standard normal distribution tables (or software) to find the area under the curve that corresponds to the event. In this case, we need to calculate the probability that a randomly selected carton weighs more than 10.21 ounces by finding the area to the right of the calculated Z-score.
Recommended video:
Guided course
07:09
Probability From Given Z-Scores - TI-84 (CE) Calculator
Related Practice
Textbook Question

Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.


Advancing Research In a survey of U.S. adults, 77% said are willing to share their personal health information to advance medical research. You randomly select 500 U.S. adults. Find the probability that the number who are willing to share their personal health information to advance medical research is (a) at most 400

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

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

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.

Textbook Question

Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.


Athletes on Social Issues In a survey of college athletes, 84% said they are willing to speak up and be more active in social issues. You randomly select 25 college athletes. Find the probability that the number who are willing to speak up and be more active in social issues is (a) at least 24