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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1:20 minutes

Problem 5.3.15

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.

P91

Verified step by step guidance

Understand the problem: The goal is to find the z-score that corresponds to the 91st percentile (P91). This means we are looking for the z-score where 91% of the data lies below it in a standard normal distribution.

Recall that the standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1. The cumulative area under the curve to the left of the z-score represents the percentile.

Use the Standard Normal Table (Z-table) or technology to find the z-score. Locate the cumulative area closest to 0.91 in the Z-table. The corresponding z-score is the value you are looking for.

If using technology (e.g., a calculator or statistical software), use the inverse cumulative distribution function (often denoted as invNorm or similar). Input the cumulative area (0.91), the mean (0), and the standard deviation (1) to calculate the z-score.

Interpret the result: The z-score you find represents the number of standard deviations above the mean where the 91st percentile lies in the standard normal distribution.

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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 indicates how many standard deviations an element is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below. Z-scores are essential for standardizing scores on different scales, allowing for comparison across different datasets.

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 is used to determine probabilities and percentiles for normally distributed data. The area under the curve corresponds to probabilities, making it a fundamental concept in statistics for understanding how data is distributed.

Cumulative Area

Cumulative area refers to the total area under the curve of a probability distribution up to a certain point. In the context of the standard normal distribution, it represents the probability that a randomly selected score will fall below a specific z-score. This concept is crucial for interpreting z-scores in terms of percentiles, allowing statisticians to understand the relative standing of a score within a distribution.

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Textbook Question

Life Spans of Fruit Flies The life spans of a species of fruit fly have a bell-shaped distribution, with a mean of 33 days and a standard deviation of 4 days.

b. The life spans of three randomly selected fruit flies are 29 days, 41 days, and 25 days. Using the Empirical Rule, find the percentile that corresponds to each life span.

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Textbook Question

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(z < - 1.11)

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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.

P33

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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.

P1.5

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Textbook Question

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Textbook Question

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Textbook Question

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.