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 5.2.12a

Chapter 5, Problem 5.2.12a

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.

Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 17 minutes.

Verified step by step guidance

Step 1: Identify the key parameters of the normal distribution. The problem states that the mean (μ) is 20 minutes and the standard deviation (σ) is 5 minutes. The random variable X represents the time an athlete uses the stairclimber.

Step 2: Standardize the value of 17 minutes to a z-score using the z-score formula: z = (X - μ) / σ. Substitute X = 17, μ = 20, and σ = 5 into the formula.

Step 3: After calculating the z-score, use the standard normal distribution table (or technology) to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the probability that the time is less than 17 minutes.

Step 4: Interpret the cumulative probability. Since the problem asks for the probability of 'less than 17 minutes,' the cumulative probability directly gives the answer.

Step 5: If using technology, input the mean, standard deviation, and the value of 17 into a statistical software or calculator to find the probability directly. Ensure the settings are for a normal distribution.

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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 continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the time spent on a stairclimber follows this distribution, indicating that most athletes will use it around the mean time, with fewer athletes using it for significantly shorter or longer durations.

Mean and Standard Deviation

The mean is the average value of a dataset, while the standard deviation measures the amount of variation or dispersion from the mean. In this scenario, the mean workout time is 20 minutes, and the standard deviation of 5 minutes indicates how much individual workout times typically deviate from this average.

Z-Scores

A Z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the value and dividing by the standard deviation. For this problem, calculating the Z-score for 17 minutes will help determine the probability of an athlete using the stairclimber for less than that time by referencing standard normal distribution tables or software.

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Related Practice

Textbook Question

Red Blood Cell Count Use the normal distribution in Exercise 16.

a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?

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.

Social Media A survey of Americans found that 55% would be disappointed if Facebook disappeared. You randomly select 500 Americans and ask them whether they would be disappointed if Facebook disappeared. Find the probability that the number who say yes is (a) less than 250

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Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Advanced Dental Admission Test The Advanced Dental Admission Test (ADAT) is designed so that the scores fit a normal distribution, as shown in the figure. (Source: American Dental Association)

b. Between what two values does the middle 50% of the ADAT scores lie?

Textbook Question

SAT Total Scores Use the normal distribution in Exercise 13.

b. Out of 1000 randomly selected SAT total scores, about how many would you expect to be greater than 1100?

Textbook Question

Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b (a<b), where (a ≤ x ≤ b) and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below.

The probability density function of a uniform distribution is

on the interval from (x=a) to (x=b). For any value of x less than a or greater than b, y=0 . In Exercises 59 and 60, use this information.

For two values c and d, where a ≤ c < d ≤ b, the probability that x lies between c and d is equal to the area under the curve between c and d, as shown below.

So, the area of the red region equals the probability that x lies between c and d. For a uniform distribution from (a=1) to (b=25) , find the probability that

a. x lies between 2 and 8.

Textbook Question

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