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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1:21 minutes

Problem 5.RE.5

Textbook Question

In Exercises 5 and 6, find the area of the indicated region under the standard normal curve. If convenient, use technology to find the area.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the area under the standard normal curve to the left of z = 0.46. This represents the cumulative probability for z-values less than or equal to 0.46.

Step 2: Recall that the standard normal curve is symmetric about z = 0, with a mean of 0 and a standard deviation of 1. The area under the curve represents probabilities.

Step 3: Use the standard normal table (z-table) or technology (such as a graphing calculator or statistical software) to find the cumulative probability corresponding to z = 0.46. The z-table provides the area to the left of a given z-value.

Step 4: If using technology, input the z-value (0.46) into the software or calculator to compute the cumulative probability. Ensure the tool is set to standard normal distribution.

Step 5: Interpret the result. The area under the curve to the left of z = 0.46 represents the probability of a random variable from a standard normal distribution being less than or equal to 0.46.

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Key Concepts

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

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve, and all values are expressed in terms of z-scores, which indicate how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve.

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. Z-scores allow for the comparison of scores from different distributions and are essential for finding areas under the standard normal curve.

Area Under the Curve

The area under the curve (AUC) in a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be calculated using z-scores and is often found using statistical tables or technology. The AUC is fundamental in hypothesis testing and confidence interval estimation.

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Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z = -1.5 and to the right of z = 1.5

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Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

Between z = -1.55 and z = 1.04

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Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

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Textbook Question

In Exercises 7–18, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

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Textbook Question

In Exercises 1 and 2, use the normal curve to estimate the mean and standard deviation.

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Textbook Question

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