Question

Elementary Statistics.. Standardized test statistic z

Answer 1

Looking for help with part C. This is my first time with this problem so im looking for step by step help.

Answer 2

Still looking for help

Answer 3

add me

Answer 4

I only fan people who help me on questions or are mods

Answer 5

Do you have the mean for each sample?

Answer 6

Um no?

Answer 7

Well you're going to need them to proceed. The \(z\) statistic is given by
\[z=\frac{\hat\theta-\theta_0}{\sigma_{\hat\theta}}\]where \(\hat\theta\) is the statistic you're estimating, \(\theta_0\) is the corresponding statistic under the null hypothesis, and \(\sigma_{\hat\theta}\) is the standard error.

The \(\hat\theta\) that you are estimating is the difference between the sample means,
\[\hat\theta=\mu_1-\mu_2\]The alternative hypothesis proposes that the first sample (kids from 1981) has a mean that is smaller than the more recent sample, so \(H_a:\mu_1<\mu_2\), or equivalently, \(H_a:\mu_1-\mu_2<0\), while the null hypothesis states the opposite, that \(H_0:\mu_1\ge\mu_2\), or \(H_0:\mu_1-\mu_2\ge0\).

This is a one-tailed test, so the test-statistic will look like
\[z=\frac{(\mu_1-\mu_2)-0}{\sigma_{\mu_1-\mu_2}}=\frac{\mu_1-\mu_2}{\sqrt{\dfrac{{\sigma_1}^2}{{n_1}^2}+\dfrac{{\sigma_2}^2}{{n_2}^2}}}\]where \(\sigma_1\) and \(\sigma_2\) are the given standard deviations of the corresponding samples, and \(n_1,n_2\) are the samples' sizes.

It doesn't look like you're given the means explicitly, and since you don't have them, what you need to do is compute the sample means.

Answer 8

Okay. I think I understand so far. Only thing is the formal im given is slightly different then the one you put...

The formal im given is shown under chapter 8.

Answer 9

My formula is exactly the same as the first one listed in that section... Are you sure this is a \(z\) test and not a \(t\) test?

Answer 10

Yours does not contain x's and you have a 0.. Yes. If you look at the question again it says " C. Find the standardized test statistic z."

Answer 11

It doesn't have to have the exact same characters to be identical. The point is that my \(\mu_1-\mu_2\) corresponds to your \(\bar x_1-\bar x_2\), while the \(\mu_1-\mu_2\) in your formula sheet corresponds to the difference between means according to the null hypothesis. When conducting a one-tailed test, this number is set to whatever value of \(k\) is given in the null hypothesis that \(\mu_1-\mu_2~\square ~k\) (\(\square\) represents any of \(\lt,\le,\gt,\ge\)).

The reason I ask if you're sure that you're conducting a \(z\) test is that the sample sizes seems rather small (probably around 20 would be my guess).

Answer 12

Okay.. Its what it says