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I'm given two random sample datasets of sample size n=20, where the first dataset represents the weights of random boys, and second group represents weights of random girls. I need to find the significance level of saying that boys on average weight more than girls.
To solve this problem I'm assuming that the null hypothesis of saying that boys weight the same as girls (no difference) is true, and building a sampling distribution of differences of weights between boys and girls with the mean difference of 0 (since I'm assuming that there is no difference in general).
I've calculated the standard deviation for this sampling distribution and I know the difference of mean weights of two sample datasets (which is supposedly a part of this sampling distribution).
Now if I divide the difference of sample means by standard deviation of this sampling distribution what value do I get? Is that a t-value? How do I find the significance level from that?

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    $\begingroup$ You do not find the significance level, $\alpha$, you choose it, ostensibly so that it reflects your willingness to make a Type I error (i.e. to reject the null hypothesis when the null hypothesis is true). One the other hand the $p$-value measures how unlikely your test statistic, $t$, is to be observed assuming that the null hypothesis is true. If $p < \alpha/2$ ($\alpha/2$ because two-sided test), then you will reject the null hypothesis at the $\alpha$ level of significance, and conclude that population mean weights in boys do not equal population mean weights in girls. $\endgroup$ – Alexis Aug 5 '14 at 2:18
  • $\begingroup$ @Alexis I'm confused between the p-value and sigma. Is the sigma a value from the sampling distribution and the p-value is the probability of getting that value? $\endgroup$ – Acute Aug 5 '14 at 4:50
  • $\begingroup$ Can you provide a citation of the use of $\sigma$ you want to understand better? The term gets used in several different ways. $\endgroup$ – Alexis Aug 5 '14 at 5:35
  • $\begingroup$ I want to understand the sigma in the context of significance level. In some places the p-value and in other places sigma. $\endgroup$ – Acute Aug 5 '14 at 8:26
  • $\begingroup$ PLease provide a citation or link using the term in a way that you do not understand. Again: $\sigma$ gets used in several ways, so such a citation or link will help prospective answerers speak to your interests. $\endgroup$ – Alexis Aug 5 '14 at 17:03
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Welch's approximate $T$ is given by: $$T = \frac{(\bar{X}_1-\bar{X}_2) - (\mu_1-\mu_2)}{\sqrt{\frac{S^2_1}{n_1}+\frac{S^2_2}{n_2}}}. $$ $T$ has an approximate $T$ distribution with $\nu^*$ degrees of freedom, where $$\nu^* = \frac{\left( \frac{S^2_1}{n_1}+\frac{S^2_2}{n_2}\right)^2} {\frac{S^4_1}{n^2_1(n_1-1)} + \frac{S^4_2}{n^2_2(n_2-1)}}.$$ It isn't immediately apparent, but is true that $\min (n_1, n_2)-1 \leq \nu^* \leq n_1+n_2-2$. In your circumstance, where the sample sizes are equal, the pooled $T$ test is rather robust to distributional assumptions. You won't go far wrong with the pooled $T$ test.

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