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I'm just doing the exercises from Stock&Watson "Introduction to econometrics". $3.12$ is about comparing men and women salaries

mean for men: $8200,$ SD $= 450, n_1=120$

mean for women: $7900,$ SD $=520, n_2=150$

I took $t= \dfrac{X_1-X_2}{\frac{\text{spooled}}{0.5(n_1+n_2)}}$

I obtained $t=5.03$ I know I have to reject H$0.$ They're asking me to calculate $p$-value. I looked for it in the tables. There is till $2.9.$ How to calculate $p$-value in that case? They're asking if the company is guilty of gender discrimination in its compensation policies. If I rejected H$0$ should I claim they are guilty?

Simon Fraser
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Emilia
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    You don't need the exact probability because all that matters is that it is so small that it is clearly smaller than the significance level you are testing at. Also, the TI-84 calculator could calculate the exact value for you if you look up how to do statistical tests on the TI-84. – Soham Konar Jan 02 '20 at 15:29

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The number of degrees of freedom is so high that your $t$-test is effectively a $z$-test. You can get an approximate $p$-value for a two-sided hypothesis by simply looking up your test statistic in a $z$-table.

Note that I chose to employ the Welch $t$-test rather than the two-sample $t$-test with the assumption of equal variances as you seem to have done.* Thus, the value of my test statistic is $$T \approx 5.07806, \quad df \approx 509.038,$$ which if we use the $z$-table, gives $$p \approx 3.813 \times 10^{-7},$$ whereas using the exact $t$ distribution with $df = 509.038$, would give instead $$p \approx 5.35858 \times 10^{-7}.$$ (Again, all $p$-values are calculated for a two-sided test.)

*Note. I should also point out that your computation of the test statistic, even for an independent two-sample $t$-test with equal variances, appears to be incorrect. I have $$T = \frac{\bar x_1 - \bar x_2}{\sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \approx 4.99739.$$ Your formula does not specify how you compute the pooled standard deviation, and I cannot replicate your result based on the formula you stated.

heropup
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