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 26m
11. Correlation1h 6m
12. Regression1h 35m
13. Chi-Square Tests & Goodness of Fit1h 57m
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 5.2.17b

Textbook Question

SAT Total Scores Use the normal distribution in Exercise 13.
b. Out of 1000 randomly selected SAT total scores, about how many would you expect to be greater than 1100?

Verified step by step guidance

Identify the key components of the problem: The SAT total scores are normally distributed, and we are tasked with finding the number of scores greater than 1100 out of 1000 randomly selected scores. This requires calculating the probability of a score being greater than 1100 and then scaling it to the sample size of 1000.

Standardize the score of 1100 using the z-score formula:

z = \frac{X - μ}{σ}

, where

X

is the score (1100),

μ

is the mean of the distribution, and

σ

is the standard deviation. These values should be provided in the problem or assumed based on typical SAT score distributions (e.g.,

μ = 1000

and

σ = 200

Use the z-score to find the cumulative probability from the standard normal distribution table or a statistical software. This gives the probability of a score being less than 1100. Subtract this value from 1 to find the probability of a score being greater than 1100:

P (X > 1100) = 1 - P (Z ≤ z)

Multiply the probability of a score being greater than 1100 by the total number of scores (1000) to find the expected number of scores greater than 1100:

E = P (X > 1100) × 1000

Interpret the result: The final value represents the expected number of SAT scores greater than 1100 out of the 1000 randomly selected scores.

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Key Concepts

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

Normal Distribution

The normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean (average) and the standard deviation (which measures the spread of the data). Understanding this distribution is crucial for making inferences about populations based on sample data.

Z-scores

A Z-score indicates how many standard deviations an element is from the mean of a distribution. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are essential for determining the probability of a score occurring within a normal distribution, allowing us to find the proportion of scores that fall above or below a certain value, such as 1100 in this case.

Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in estimating the likelihood of scores exceeding a certain threshold, such as 1100, by providing a quick way to assess how many scores lie beyond a specified number of standard deviations from the mean.

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Textbook Question

Finding Area

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

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Textbook Question

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

Between z= -1.55 and z= 1.55

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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(x > 182)

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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)

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a. What percent of the new mothers had a pregnancy length of less than 290 days?

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Red Blood Cell Count Use the normal distribution in Exercise 16.

a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?

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Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Weights of Teenagers In a survey of 18-year old males, the mean weight was 166.7 pounds with a standard deviation of 49.3 pounds. (Adapted from National Center for Health Statistics)

c. What weight represents the first quartile?

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Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

COVID-19 Response Surveyors asked respondents to rate ten key aspects of their government’s response to the COVID-19 pandemic, including preparedness, communication, and material aid. A pandemic response score that ranged from 0 to 100 was calculated. The mean score for U.S. respondents was 50.6 with a standard deviation of 29.0. (Source: PLOS One)

b. What score represents the 61st percentile?

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SAT Total Scores Use the normal distribution in Exercise 13.b. Out of 1000 randomly selected SAT total scores, about how many would you expect to be greater than 1100?

Key Concepts

Normal Distribution

Z-scores

Empirical Rule

Watch next

SAT Total Scores Use the normal distribution in Exercise 13.
b. Out of 1000 randomly selected SAT total scores, about how many would you expect to be greater than 1100?