Math  /  Data & Statistics

Question1
A population has mean μ=21\mu=21 and standard deviation σ=6\sigma=6. Find μxˉ\mu_{\bar{x}} and σxˉ\sigma_{\bar{x}} for samples of size n=25n=25. Round your answers to one decimal place if needed. μxˉ=σxˉ=\begin{array}{l} \mu_{\bar{x}}=\square \\ \sigma_{\bar{x}}=\square \end{array} \square

Studdy Solution

STEP 1

What is this asking? Given a population's mean and standard deviation, what are the mean and standard deviation of the *means* of samples of size 25? Watch out! Don't mix up population parameters and sample statistics!
We're dealing with the distribution of sample *means* here.

STEP 2

1. Calculate the mean of the sample means.
2. Calculate the standard deviation of the sample means.

STEP 3

The mean of the sample means, denoted by μxˉ\mu_{\bar{x}}, is simply equal to the population mean, μ\mu.
This is a key concept in statistics!
It means that if we take tons of samples and calculate their means, the average of *those* means will be the same as the original population mean.
Pretty cool, right?

STEP 4

In this case, the population mean is given as μ=21\mu = 21.
Therefore, μxˉ=μ=21\mu_{\bar{x}} = \mu = 21.

STEP 5

The standard deviation of the sample means, also known as the **standard error**, is denoted by σxˉ\sigma_{\bar{x}}.
It's calculated using this **super important** formula: σxˉ=σn \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} where σ\sigma is the population standard deviation and nn is the sample size.
This formula tells us how spread out the sample means are.

STEP 6

We're given that σ=6\sigma = 6 and n=25n = 25.
Let's **plug those values** into the formula: σxˉ=625 \sigma_{\bar{x}} = \frac{6}{\sqrt{25}}

STEP 7

Now, we **simplify**: σxˉ=65 \sigma_{\bar{x}} = \frac{6}{5}

STEP 8

And finally, we **calculate the result**: σxˉ=1.2 \sigma_{\bar{x}} = 1.2

STEP 9

So, we found that μxˉ=21\mu_{\bar{x}} = 21 and σxˉ=1.2\sigma_{\bar{x}} = 1.2.
That means the mean of the sample means is **21**, and the standard deviation of the sample means is **1.2**!

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