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 5.1.39

Chapter 5, Problem 5.1.39

Computing and Interpreting z-Scores In Exercises 39 and 40, (a) find the z-score that corresponds to each value and (b) determine whether any of the values are unusual.

Stanford-Binet IQ Scores The test scores for the Stanford-Binet Intelligence Scale are normally distributed with a mean score of 100 and a standard deviation of 16. The test scores of four students selected at random are 98, 65, 106, and 124.

Verified step by step guidance

Step 1: Recall the formula for calculating a z-score:

z = \frac{x - μ}{σ}

, where

x

is the data value,

μ

is the mean, and

σ

is the standard deviation.

Step 2: Substitute the given values into the formula for each test score. The mean

μ

is 100, and the standard deviation

σ

is 16. For example, for the first score of 98, calculate

z = \frac{98 - 100}{16}

Step 3: Repeat the calculation for the other scores: 65, 106, and 124. For each score, use the formula

z = \frac{x - μ}{σ}

and substitute the respective values.

Step 4: Determine whether any of the z-scores are unusual. A z-score is considered unusual if it is less than -2 or greater than 2. Compare each calculated z-score to this threshold.

Step 5: Interpret the results. For any z-scores that are unusual, explain what this means in the context of the problem (e.g., the corresponding test score is significantly different from the mean).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. A z-score indicates how many standard deviations an element is from the mean, allowing for comparison across different datasets.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three, which is known as the empirical rule.

Unusual Values

In statistics, a value is considered unusual if it lies more than two standard deviations away from the mean in a normal distribution. This threshold helps identify outliers or extreme values that may warrant further investigation, as they can significantly impact the results and interpretations of the data.

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Related Practice

Textbook Question

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In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

Between z=0 and z=2.86

Textbook Question

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Textbook Question

Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.

For a random sample of n=36, find the probability of a sample mean being less than 12,750 or greater than 12,753 when mu=12750 and 1.7.

Textbook Question

Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

Salaries The annual salaries for web software development managers are normally distributed, with a mean of about \$136,000 and a standard deviation of about \$11,500. Random samples of 40 are drawn from this population, and the mean of each sample is determined.

Textbook Question

Graphical Analysis In Exercises 9 and 10, the graph of a population distribution is shown with its mean and standard deviation. Random samples of size 100 are drawn from the population. Determine which of the figures labeled (a)–(c) would most closely resemble the sampling distribution of sample means. Explain your reasoning.

The waiting time (in seconds) to turn left at an intersection

Textbook Question

Computing Probabilities for Normal Distributions In Exercises 1–6, the random variable x is normally distributed with mean mu=174 and standard deviation sigma=20. Find the indicated probability.

P(172 < x <192)