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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2:03 minutes

Problem 5.1.39

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.

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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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

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

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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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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.

0.6736

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