Textbook Question
Finding Area In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z= -1.28 and to the right of z= 1.28
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 5.4.2
In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.
Mu = 45, sigma =15, n = 100
Verified step by step guidance1
Step 1: Recall the formula for the mean of the sampling distribution of sample means. The mean of the sampling distribution (denoted as μₓ̄) is equal to the population mean (μ). Therefore, μₓ̄ = μ.
Step 2: Substitute the given value of the population mean (μ = 45) into the formula. This means that the mean of the sampling distribution of sample means is also 45.
Step 3: Recall the formula for the standard deviation of the sampling distribution of sample means, also known as the standard error (SE). The formula is SE = σ / √n, where σ is the population standard deviation and n is the sample size.
Step 4: Substitute the given values into the formula for SE. Here, σ = 15 and n = 100. The formula becomes SE = 15 / √100.
Step 5: Simplify the expression for SE by calculating the square root of the sample size (√100 = 10) and dividing the population standard deviation by this value. This will give you the standard deviation of the sampling distribution of sample means.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution
The sampling distribution of the sample mean is the probability distribution of all possible sample means from a population. It describes how the sample means vary from sample to sample and is crucial for understanding the behavior of sample statistics. According to the Central Limit Theorem, as the sample size increases, the sampling distribution approaches a normal distribution, regardless of the population's distribution.
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Sampling Distribution of Sample Proportion
Mean of the Sampling Distribution
The mean of the sampling distribution, also known as the expected value of the sample mean, is equal to the population mean (mu). This means that if you take many samples from a population and calculate their means, the average of those sample means will converge to the population mean. In this case, with mu = 45, the mean of the sampling distribution is also 45.
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05:11Sampling Distribution of Sample Proportion
Standard Deviation of the Sampling Distribution (Standard Error)
The standard deviation of the sampling distribution, often referred to as the standard error, measures the dispersion of sample means around the population mean. It is calculated by dividing the population standard deviation (sigma) by the square root of the sample size (n). For this problem, with sigma = 15 and n = 100, the standard error would be 15 / √100 = 1.5.
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08:45Calculating Standard Deviation
Related Practice
Textbook Question
Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.
For a random sample of n=45, find the probability of a sample mean being greater than 551 when mu=550 and sigma=3.7.
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.
0.6736
Textbook Question
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.
Textbook Question
In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.
P(x > 65)
1
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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(- 0.89 < z < 0)