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

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

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1:09 minutes

Problem 5.4.2

Textbook Question

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 guidance

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.

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.

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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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.

As the sample size increases, the standard deviation of the distribution of sample means increases.

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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.

As the sample size increases, the mean of the distribution of sample means increases.

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Textbook Question

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.

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Textbook Question

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Textbook Question

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.

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For a random sample of n=64, find the probability of a sample mean being less than 24.3 when Mu=24 and sigma=1.25.

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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

Key Concepts

Sampling Distribution

Mean of the Sampling Distribution

Standard Deviation of the Sampling Distribution (Standard Error)

Watch next

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