Math Snap

STEP 1

What is this asking?
We're checking if the true average is less than 20, given a sample average of 18 from 48 data points and a population standard deviation of 4.6.
Watch out!
Don't mix up the sample and population standard deviations!
We're given the population standard deviation here, so we'll use a z-test, not a t-test.

STEP 2

1. Set up the z-test
2. Calculate the test statistic

STEP 3

Alright, so we're dealing with a hypothesis test here!
Our null hypothesis $H_0$ says the population mean $\mu$ is 20.
But we suspect it might actually be less than 20, and that's our alternative hypothesis $H_a$.

STEP 4

Since we know the population standard deviation $\sigma = 4.6$, we're going to use a z-test.
The z-test formula is: $z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$.
This formula tells us how far our sample mean $\overline{X}$ is from the hypothesized population mean $\mu$, measured in terms of standard errors.

STEP 5

Let's plug in what we know.
We have $\overline{X} = 18$, $\mu = 20$ (from our null hypothesis), $\sigma = 4.6$, and $n = 48$.

STEP 6

Substituting the values into our formula, we get: $z = \frac{18 - 20}{\frac{4.6}{\sqrt{48}}}$.

STEP 7

Let's simplify the denominator first: $\frac{4.6}{\sqrt{48}} \approx \frac{4.6}{6.928} \approx 0.664$.
So, our standard error is approximately 0.664.

STEP 8

Now, let's compute the numerator: $18 - 20 = -2$.
This tells us our sample mean is 2 units below our hypothesized mean.

STEP 9

Finally, divide the numerator by the denominator: $z = \frac{-2}{0.664} \approx -3.01$.
So, our test statistic is approximately $-3.01$.

Our test statistic is approximately $-3.01$.
This means our sample mean is about 3 standard deviations below the hypothesized population mean.