Question

A population has a standard deviation of 50. It is sampled with samples of size $100 .$ What is the variance of the means of the samples?

   A population has a standard deviation of 50. It is sampled with samples of size $100 .$ What is the variance of the means
of the samples?

Introductory Business Statistics

Introductory Business Statistics

Alexander Holmes,… 1st Edition

Chapter 7, Problem 64

Instant Answer

Solved by Expert Willis James

09/17/2023

Step 1

Step 1:** Calculate the sample standard deviation using the formula: sample standard deviation = population standard deviation / sqrt(sample size) Given that the population standard deviation is 50 and the sample size is 100: sample standard deviation = 50 / Show more…

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A population has a standard deviation of 50. It is sampled with samples of size $100 .$ What is the variance of the means of the samples?

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Key Concepts

Sampling Distribution of the Mean

This concept refers to the distribution of sample means obtained by repeatedly drawing samples from a population. It explains that although individual observations have their own variance, the average of samples will tend to have a smaller variance, specifically the population variance divided by the sample size.

Standard Error

The standard error is the measure of variability of the sample mean and is calculated as the population standard deviation divided by the square root of the sample size. It quantifies how much the sample mean is expected to fluctuate from the true population mean across samples.

Variance Reduction with Increased Sample Size

When calculating the variance of the sample means, the original population variance is divided by the sample size. This reflects the principle that larger samples provide more precise estimates of the population mean, thereby reducing the variance or dispersion of the sample means.

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