Find the critical value(s) for the alternative hypothesis, lev...

Question

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal and (b) not equal.
Ha:μ1>μ2 , α=0.01 , n1=12 , n2=15

Ha:μ1>μ2 , α=0.01 , n1=12 , n2=15

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the following test scenario, calculate the critical value or values for both I equal variances and Ii not equal variances. Assume random independent samples from normal populations. H1 is greater than 2, alpha is equal to 0.02, N1 is equal to 16, N2 is equal to 11. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. We're asked to solve for the critical value or values for both our first answer equal variances, and our second answer not equal variances, given the information provided to us. So we're going to use the information that is provided to us to help us solve for equal variances and not equal variances for this particular problem. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers and to see what our final answer pair might be. Noting that I'm going to read the answer for part I first and then the answer for I I second. So A is 2.145 and 2.539. B is 2.485 and 2.681. C is 2.110 and 2.430. And D is 2.485 and 2.539. Awesome. So, First off, our first step that we need to take to solve this problem is we need to recall and note that this is a right tail to sample T test because we are told that HA, which denotes the alternative hypothesis, states that mu 1 is greater than mu 2. Our second step now. And we're focusing on solving for part I. For equal variances are degrees of freedom, which will denote as DF is equal to N1 plus N2 minus 2, which is equal to we plug in our known variables. 16 + 11 minus 2, which is equal to 25. Fantastic. So our third step now. we need to find the critical value for our significance level or level of significance for alpha equals 0.02, and we're also told once again that our degrees of freedom is equal to 25, and we're also told that this is a right-tailed test which I'll call RT for short. So that would mean that T subscript 0.02.25 will be approximately equal to 2.485. Fantastic. So that is our first answer for part I. Hooray, we did it. So now we need to solve for part II. So now for our fourth step, solving for part II, our second answer for not equal variances, we need to approximate the degrees of freedom, which the degrees of freedom is approximately equal to the minimum of. N1 minus 1. N2 minus 1, which will mean that this is equal to the minimum of, so the minimum of 15.10, which is equal to simply 10. Fantastic. So now, moving right along here, our 5th step is we need to find the critical value for Our significance level or level of significance of 0.02 and the degrees of freedom of 10, so let's make a note of this. 0.02 is equal to alpha, and our degrees of freedom for this particular case is equal to 10, and once again we're dealing with a right-tailed test. So that will mean for this particular case, T subscript 0.02.10 will be approximately equal to 2.539, and that is our second answer for part II. Hooray, we did it. So, that will mean That without a doubt when we round to three decimal places, our final two answers once again, which corresponds with multiple choice answer letter D, will have to be for part I 2.485 and for II 2.539. And that's it. Those are our final two answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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