The critical value of z can tell what probability any particular variable will have. Z critical value is a point that cuts off an area under the standard normal distribution. The critical value of t helps to decide if a null hypothesis should be supported or rejected. T value is used in a hypothesis test to compare against a calculated t score. Again, with our decision of a 5% risk, we can reject the null hypothesis of equal mean body fat for men and women.T critical value is a point that cuts off the student t distribution. You can see that the degrees of freedom are 20.9888. As was mentioned above, this test also has a complex formula for degrees of freedom. This test does not use the pooled estimate of the standard deviation. The figure also shows the results for the t-test that does not assume equal variances. It is important to make this decision before doing the statistical test. We decided on a 5% risk of concluding the mean body fat for men and women are different, when they are not. We can reject the hypothesis of equal mean body fat for the two groups and conclude that we have evidence body fat differs in the population between men and women. The one-sided tests are for one-sided alternative hypotheses – for example, for a null hypothesis that mean body fat for men is less than that for women. Our alternative hypothesis is that the mean body fat is not equal. Our null hypothesis is that the mean body fat for men and women is equal. The two-sided test is what we want (Prob > |t|). The software shows results for a two-sided test and for one-sided tests. The results for the two-sample t-test that assumes equal variances are the same as our calculations earlier. The estimate adjusts for different group sizes. This builds a combined estimate of the overall standard deviation. Next, we calculate the pooled standard deviation. This difference in our samples estimates the difference between the population means for the two groups. This calculation begins with finding the difference between the two averages: We start by calculating our test statistic. We'll further explain the principles underlying the two sample t-test in the statistical details section below, but let's first proceed through the steps from beginning to end. But how different are they? Are the averages “close enough” for us to conclude that mean body fat is the same for the larger population of men and women at the gym? Or are the averages too different for us to make this conclusion? Without doing any testing, we can see that the averages for men and women in our samples are not the same. We want to know if we have evidence that the mean grams of protein for the two brands of energy bars is different or not. Our idea is that the mean grams of protein for the underlying populations for the two brands may be different. Our measurement is the grams of protein for each energy bar.
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