Question

A population has a mean of 50 and a standard deviation of $6 .$ (a) What are the mean and standard deviation of the sampling distribution of the mean for $\mathrm{N}=$ $16 ?$ (b) What are the mean and standard deviation of the sampling distribution of the mean for $\mathrm{N}=20 ?$

Name: Problem 1, Chapter 9: Introduction to Statistics
Uploaded: 2023-09-25T07:35:29-08:00
Duration: 1 min 54 s
Channel: Anna D.
Description: A population has a mean of 50 and a standard deviation of 6 . (a) What are the mean and standard deviation of the sampling distribution of the mean for N= 16 ? (b) What are the mean and standard deviation of the sampling distribution of the mean for N=20 ?

   A population has a mean of 50 and a standard deviation of $6 .$ (a) What are the mean and standard deviation of the sampling distribution of the mean for $\mathrm{N}=$ $16 ?$ (b) What are the mean and standard deviation of the sampling distribution of the mean for $\mathrm{N}=20 ?$

Introduction to Statistics

David Lane 2003 Edition

Chapter 9, Problem 1

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Solved by Expert Anna D.

09/25/2023

Step 1

The standard deviation of the sampling distribution of the mean (also called the standard error) is calculated as the population standard deviation divided by the square root of the sample size: Standard Error (SE) = σ / √N SE = 6 / √16 SE = 6 / 4 SE = 1.5 So, Show more…

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A population has a mean of 50 and a standard deviation of $6 .$ (a) What are the mean and standard deviation of the sampling distribution of the mean for $\mathrm{N}=$ $16 ?$ (b) What are the mean and standard deviation of the sampling distribution of the mean for $\mathrm{N}=20 ?$

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Key Concepts

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of sample means over all possible samples of a certain size taken from a population. It is central to inferential statistics because, according to the Central Limit Theorem, for sufficiently large samples, the distribution of the sample mean will approximate a normal distribution regardless of the original population’s distribution. The mean of this sampling distribution is equal to the population mean, making it an unbiased estimator of the population mean.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It measures the extent to which the sample mean is expected to fluctuate from the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size, which shows that increasing the sample size will decrease the sampling variability of the sample mean.

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