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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2:27 minutes

Problem 5.1.45

Textbook Question

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.

Verified step by step guidance

Step 1: Identify the problem. The question asks for the probability of z occurring in the shaded region of the standard normal distribution. From the image, the shaded region lies between z = -1 and z = 1.

Step 2: Understand the standard normal distribution. It is a bell-shaped curve with a mean of 0 and a standard deviation of 1. Probabilities are calculated using the area under the curve.

Step 3: Use the z-scores provided (-1 and 1) to find the cumulative probabilities. The cumulative probability for a z-score represents the area under the curve to the left of that z-score.

Step 4: To find the probability of z occurring in the shaded region, calculate the cumulative probability for z = 1 and subtract the cumulative probability for z = -1. This gives the area between these two z-scores.

Step 5: If convenient, use technology (such as a graphing calculator, statistical software, or online tools) to find the cumulative probabilities for z = -1 and z = 1. Subtract the smaller cumulative probability from the larger one to determine the probability of z occurring in the shaded region.

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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, which is symmetric about the mean. This distribution is crucial for calculating probabilities associated with z-scores, which indicate how many standard deviations an element is from the mean.

Z-scores

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 probabilities in the standard normal distribution.

Probability Area

The probability area under the curve of the standard normal distribution represents the likelihood of a random variable falling within a specific range of values. In the context of z-scores, the shaded region between two z-scores indicates the probability of a value falling between those scores. This area can be calculated using statistical tables or technology, such as calculators or software.

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