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

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

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2:21 minutes

Problem 5.Q.5

Textbook Question

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)

What percent of the IQ scores are greater than 112?

Verified step by step guidance

Identify the key parameters of the normal distribution: the mean (μ = 100) and the standard deviation (σ = 15). The problem asks for the percentage of IQ scores greater than 112.

Standardize the raw score of 112 to a z-score using the formula:

z = \frac{x - μ}{σ}

, where x is the raw score, μ is the mean, and σ is the standard deviation.

Substitute the values into the z-score formula:

z = \frac{112 - 100}{15}

. Simplify the numerator and divide by the standard deviation to calculate the z-score.

Use a z-table or a standard normal distribution calculator to find the cumulative probability corresponding to the calculated z-score. This gives the probability of a score being less than 112.

To find the percentage of scores greater than 112, subtract the cumulative probability from 1 (i.e.,

1 - P

, where P is the cumulative probability). Multiply the result by 100 to express it as a percentage.

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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. In the context of IQ scores, this means that most individuals score around the average (100), with fewer individuals scoring significantly higher or lower.

Z-Score

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. For an IQ score of 112, the Z-score helps determine its position relative to the mean and is essential for finding the percentage of scores above this value.

Percentile Rank

Percentile rank is a measure used to indicate the value below which a given percentage of observations fall. In this case, calculating the percentile rank for an IQ score of 112 allows us to determine the percentage of individuals who scored lower than this score, which can then be used to find the percentage of scores that are greater.

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Textbook Question

In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P85

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Textbook Question

In Exercises 61 and 62, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

A survey of U.S. adults ages 33 to 40 earning more than $150,000 per year found that 94% are content with how their lives have turned out so far. You randomly select 20 U.S. adults ages 33 to 40 earning more than $150,000 and ask if they are content with their lives so far.

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Textbook Question

In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 36)

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Textbook Question

In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x < 60)

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In a survey of U.S. adults, 81% feel they have little or no control over data collected about them by companies. You randomly select 250 U.S. adults and ask them whether they feel they have control over data collected about them by companies. Use this information in Exercises 11 and 12. (Source: Pew Research Center)

Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

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Textbook Question

Assume the machine shifts and the distribution of the amount of the compound added now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added.

a. What is the probability that you select a vial that is within the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.)

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Textbook Question

A survey of adults in the United States found that 61% ate at a restaurant at least once in the past week. You randomly select 30 adults and ask them whether they ate at a restaurant at least once in the past week. (Source: Gallup)

c. Is it unusual for exactly 14 out of 30 adults to have eaten in a restaurant at least once in the past week? Explain your reasoning.

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Textbook Question

The random variable x is normally distributed with the given parameters. Find each probability.

a. μ = 9.2, σ ≈ 1.62, P(x < 5.97)

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In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)What percent of the IQ scores are greater than 112?

Key Concepts

Normal Distribution

Z-Score

Percentile Rank

Watch next

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)

What percent of the IQ scores are greater than 112?