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 5.2.15

Chapter 5, Problem 5.2.15

Graphical Analysis In Exercises 13–16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed.

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Step 1: Identify the key parameters of the normal distribution from the graph. The mean (μ) is given as 267 days, and the standard deviation (σ) is 10 days. The shaded region corresponds to the range 285 < x < 294.

Step 2: Convert the given range (285 < x < 294) into z-scores using the formula:

z = \frac{x - μ}{σ}

. For x = 285 and x = 294, calculate the respective z-scores.

Step 3: Use a standard normal distribution table or software to find the cumulative probabilities corresponding to the calculated z-scores. The cumulative probability for z = (285 - 267)/10 and z = (294 - 267)/10 will be determined.

Step 4: Subtract the cumulative probability at z = 285 from the cumulative probability at z = 294 to find the probability of the shaded region. This gives the area under the curve between these two z-scores.

Step 5: Interpret the result as the probability that a randomly selected member of the population has a pregnancy length between 285 and 294 days.

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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 (μ) and standard deviation (σ). In this context, the pregnancy lengths are normally distributed, meaning most values cluster around the mean, with fewer values appearing as you move away from the mean in either direction.

Probability

Probability quantifies the likelihood of an event occurring, expressed as a number between 0 and 1. In this scenario, we are interested in finding the probability that a randomly selected pregnancy length falls within a specific range (285 to 294 days), which can be calculated using the area under the normal distribution curve corresponding to that interval.

Z-scores

A Z-score indicates how many standard deviations an element is from the mean of a distribution. It is calculated using the formula Z = (X - μ) / σ. In this case, Z-scores can be used to standardize the values of 285 and 294, allowing us to find the corresponding probabilities from the standard normal distribution table.

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Related Practice

Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.

P(x≥109)

a. P(x>109.5)

b. P(x<108.5)

c. P(x<109.5)

d. P(x>108.5)

Textbook Question

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.

Find the z-score that has 2.275% of the distribution’s area to its left.

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.

An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter.

Textbook Question

Using and Interpreting Concepts

Finding Area In Exercises 17–22, find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area.

Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P33

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.

If the sample size is at least 30, then you can use z-scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.