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(- 1.54 < z < 1.54)
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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 5.4.1
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 = 150, sigma =25, n = 49
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 = 150) into the formula. This means the mean of the sampling distribution is μₓ̄ = 150.
Step 3: Recall the formula for the standard deviation of the sampling distribution of sample means. The standard deviation of the sampling distribution (denoted as σₓ̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). The formula is: σₓ̄ = σ / √n.
Step 4: Substitute the given values into the formula for the standard deviation. Use σ = 25 and n = 49. This gives: σₓ̄ = 25 / √49.
Step 5: Simplify the expression for the standard deviation. Calculate the square root of 49 (which is 7) and divide 25 by 7 to find the value of σₓ̄. This will give you the standard deviation of the sampling distribution.
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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 means of different samples will vary and is crucial for understanding the behavior of sample statistics. The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution will approach a normal distribution as the sample size increases.
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Sampling Distribution of Sample Proportion
Mean of the Sampling Distribution
The mean of the sampling distribution of the sample means, also known as the expected value, is equal to the population mean (mu). This means that if you take many samples and calculate their means, the average of those means will converge to the population mean. In this case, with mu = 150, the mean of the sampling distribution will also be 150.
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Standard Deviation of the Sampling Distribution (Standard Error)
The standard deviation of the sampling distribution, often referred to as the standard error, measures the variability 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 = 25 and n = 49, the standard error can be computed to understand how much sample means will typically deviate from the population mean.
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08:45Calculating Standard Deviation
Related Practice
Textbook Question
Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.
Renewable Energy During a recent period of two years, the day-ahead prices for renewable energy in Germany (in euros per mega-watt hour) have a mean of 31.58 and a standard deviation of 12.293. Random samples of size 75 are drawn from this population, and the mean of each sample is determined.
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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 ≤ 150)
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Textbook Question
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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 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
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