Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.T.8

Chapter 5, Problem 5.T.8

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?

Verified step by step guidance

Step 1: Identify the key parameters of the normal distribution. The mean (μ) is \$44,000, and the standard deviation (σ) is \$2,450. The problem asks for the proportion of residents with disposable incomes between \$40,000 and \$42,000.

Step 2: Convert the income values (\$40,000 and \$42,000) into z-scores using the formula:

z = \frac{x - μ}{σ}

, where x is the income value, μ is the mean, and σ is the standard deviation.

Step 3: Look up the z-scores in the standard normal distribution table (or use a calculator) to find the cumulative probabilities corresponding to the z-scores for \$40,000 and \$42,000.

Step 4: Calculate the proportion of residents with incomes between \$40,000 and \$42,000 by subtracting the cumulative probability for \$40,000 from the cumulative probability for \$42,000.

Step 5: Multiply the proportion obtained in Step 4 by the total number of residents (800) to estimate the number of residents with disposable incomes in the specified range.

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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 per capita disposable income follows a normal distribution, which allows us to use statistical methods to estimate probabilities and expectations.

Z-Scores

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 are essential for determining how many standard deviations an element is from the mean, which is crucial for finding probabilities in a normal distribution.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in estimating the proportion of residents with disposable incomes within a specific range, such as between $40,000 and $42,000 in this scenario.

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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

In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6

Find each probability.

b. P(0 < x < 5)

Textbook Question

In Exercises 5 and 6, 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.

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (a) exactly 7. Identify any unusual events. Explain.

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

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.