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 = -2.825
Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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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.RE.28
In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.
P(x < 55)
Verified step by step guidance1
Step 1: Recognize that the problem involves a normal distribution with a mean (μ) of 74 and a standard deviation (σ) of 8. The goal is to find the probability P(x < 55).
Step 2: Standardize the value of x = 55 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (55 - 74) / 8.
Step 3: Simplify the z-score formula to calculate the z-value. This will give you the standardized value corresponding to x = 55.
Step 4: Use a standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to the calculated z-value. This cumulative probability represents P(x < 55).
Step 5: Interpret the result. The cumulative probability obtained from the z-table or software is the probability that the random variable x is less than 55 in the given normal distribution.
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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 continuous probability distribution characterized by its bell-shaped curve, defined by its mean (mu) and standard deviation (sigma). It is symmetric around the mean, meaning that approximately 68% of the data falls within one standard deviation from the mean, and about 95% falls within two standard deviations. This distribution is fundamental in statistics for modeling real-world phenomena.
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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 normal distributions and are essential for finding probabilities in a standard normal distribution table.
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Probability Calculation
Probability calculation in the context of a normal distribution involves determining the likelihood of a random variable falling within a certain range. This is often done using Z-scores to convert the values into a standard normal distribution, where probabilities can be easily found using statistical tables or software. For the question at hand, calculating P(x < 55) requires finding the Z-score for x = 55 and then using it to find the corresponding probability.
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Related Practice
Textbook Question
In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.
P(0.42 < z < 3.15)
Textbook Question
An auto parts seller finds that 1 in every 200 parts sold is defective. Use the geometric distribution to find the probability that (c) none of the first 20 parts sold are defective.
1
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Textbook Question
Find the positive z-score for which 94% of the distribution’s area lies between -z and z.
Textbook Question
In Exercises 33 and 34, find the indicated probabilities. If convenient, use technology to find the probabilities.
The daily surface concentration of carbonyl sulfide on the Indian Ocean is normally distributed, with a mean of 9.1 picomoles per liter and a standard deviation of 3.5 picomoles per liter. Find the probability that on a randomly selected day, the surface concentration of carbonyl sulfide on the Indian Ocean is
a. between 5.1 and 15.7 picomoles per liter.
Textbook Question
In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.
P(x ≤ 36)