Textbook Question
Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.
Find the z-score that has 2.275% of the distribution’s area to its left.
Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 5, Problem 5.1.17
Using and Interpreting Concepts
Finding Area In Exercises 17–22, find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area.
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the area of the shaded region under the standard normal curve. The shaded region represents the cumulative probability from z = -∞ to z = 0.6.
Step 2: Recall that the standard normal curve is symmetric about z = 0, and the total area under the curve is 1. The area to the left of z = 0.6 can be found using the cumulative distribution function (CDF) of the standard normal distribution.
Step 3: Use the z-score table or technology (such as a graphing calculator or statistical software) to find the cumulative probability corresponding to z = 0.6. This value represents the area under the curve to the left of z = 0.6.
Step 4: If using a z-score table, locate the row corresponding to z = 0.6. The table will provide the cumulative probability for this z-score. If using technology, input the z-score into the software's standard normal CDF function.
Step 5: Interpret the result. The cumulative probability obtained represents the proportion of the distribution that lies to the left of z = 0.6, which is the area of the shaded region.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
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 statistical analysis, particularly in hypothesis testing and confidence intervals.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table
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 are essential for determining the position of a data point within a standard normal distribution, allowing for the comparison of scores from different distributions.
Recommended video:
Guided course
06:31Z-Scores From Given Probability - TI-84 (CE) Calculator
Area Under the Curve
In the context of the normal distribution, the area under the curve represents the probability of a random variable falling within a particular range. The total area under the curve is equal to 1, and specific areas can be calculated using z-scores and standard normal distribution tables or technology. This concept is fundamental for understanding probabilities and making inferences about populations based on sample data.
Recommended video:
Guided course
08:50Z-Scores from Probabilities
Related Practice
Textbook Question
In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.
An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter.
Textbook Question
Graphical Analysis In Exercises 13–16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed.
" style="" width="282">
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
Textbook Question
True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
If the sample size is at least 30, then you can use z-scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.
Textbook Question
Graphical Analysis In Exercises 11–16, determine whether the graph could represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal distribution, estimate the mean and standard deviation.