Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Probabilities & Z-Scores w/ Graphing Calculator

Statistics

6. Normal Distribution and Continuous Random Variables

Probabilities & Z-Scores w/ Graphing Calculator

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1:19 minutes

Problem 5.1.8

Textbook Question

Explain how to transform a given x-value of a normally distributed variable x into a z-score.

Verified step by step guidance

Identify the mean (μ) and standard deviation (σ) of the normally distributed variable x. These values are essential for calculating the z-score.

Recall the formula for calculating a z-score:

z = \frac{x - μ}{σ}

. This formula standardizes the x-value by measuring how many standard deviations it is away from the mean.

Subtract the mean (μ) from the given x-value. This step calculates the deviation of x from the mean.

Divide the result from step 3 by the standard deviation (σ). This step scales the deviation by the variability of the distribution, converting it into a z-score.

Interpret the z-score: A positive z-score indicates the x-value is above the mean, while a negative z-score indicates it is below the mean. The magnitude of the z-score shows how far the x-value is from the mean in terms of standard deviations.

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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, showing 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 two parameters: the mean (average) and the standard deviation (which measures the spread of the data). Understanding this distribution is crucial for statistical analysis, as many statistical tests assume normality.

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 x-value and then dividing by the standard deviation. This standardization allows for comparison between different datasets or distributions, as it indicates how many standard deviations an element is from the mean.

Standardization Process

The standardization process involves converting a normal variable into a z-score to facilitate comparison across different scales or distributions. This is done using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. This transformation is essential in statistics for hypothesis testing and creating confidence intervals.

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Textbook Question

Building Basic Skills and Vocabulary

A student’s grade on the Fundamentals of Engineering exam has a z-score of −0.5. Make an observation about the student’s grade.

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Textbook Question

True or False? In Exercises 7–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

It is impossible to have a z-score of 0.

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Textbook Question

Graphical Analysis In Exercises 41 and 42, the midpoints A, B, and C are marked on the histograms at the left. Match them with the indicated z-scores. Which z-scores, if any, would be considered unusual?

z = 0, z = 2.14, z = −1.43

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Textbook Question

The mean annual salary for a sample of electrical engineers is $86,500, with a standard deviation of $1500. The data set has a bell-shaped distribution.

b. The salaries of three randomly selected electrical engineers are $93,500, $85,600, and $82,750. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.

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Textbook Question

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.

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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)

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Textbook Question

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.

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Textbook Question

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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