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Elementary Statistics a Step by Step Approach
5 months, 3 weeks ago
Here. We want to find the P values for four scenarios shown here. So let's take a look at the first one. The test statistic came out to -1.584. The sample size was 19. And the alternative hypothesis is that the mean is not equal to the no hypothesized mean. So this test statistic follows a T distribution with 19 -1° of freedom. So it follows a T distribution with 18° of freedom. If we want to visualize this on T distribution. So let's say this distribution is the T distribution With 18° of freedom. So every ti distribution has to be specified by the degrees of freedom. The higher the degrees of freedom, the narrower the distribution becomes. This is where our test statistic landed -1.584. So what is the p value? The key to determining how to calculate the p value is to look at the alternative hypothesis. If it says not equal to it means the p value is the probability of getting a test statistic at least as extreme as the one we got, if it says less, then then the p value is the probability of getting a test statistic less than the one we got. And similarly, if it's greater then then it's the probability of getting a T statistic greater than the one that we got. So going back to the curve here, This is -1.584 And this one is positive, 1.584. The p value is the probability of getting a test statistic At least as extreme as the one We got. So that means it's equal to the area in this tale. Below our test statistic as well as the area above positive of our test statistic. So each of these areas is the P value divided by two or half the p value. So for a P value is equal to the probability of tea being less than -1.584 plus. The probability of teaching Greater than positive 1.584. And because of the symmetry of the T distribution, these two terms are identical. So we can simply re express it as two times the probability that T is greater than 1.584. So if we use a tea table to find this probability We know that it's 18° of freedom. So we're dealing with this role And the T value is 1.584. So it's in between these two columns. So the area in the one tail above 1.584 Is between .05 and .1. And then since we are doubling that it's between .1 and .2. So that's what we know about the p value from our tea table. And this can also be solved quite easily in Excel. So I'll show you the formula to use an Excel to do this. Now the way the formula in Excel is set up, it's set up better to solve cumulative probabilities. So areas to the left of valleys than it is to solve areas to the right of valleys. So it'll be easy to solve probability The T is less than -1.584. So in Excel we simply type equals and then T dot so that's the first function we see here, then we enter our test statistic value. So this time it's minus 1.584 degrees of freedom is 18 and we this the third argument is whether or not we want to solve a cumulative probability, a cumulative probability is just all of the area to the left of this value. So yes, that is the case here. So we enter true and then hit enter And we get .065. So it's two times .065 Comes out to 0.13. And we can see that this more precise answer is an agreement with the answer we got from the tea table. Now looking at part B, it's a less than alternative hypothesis. And so the p value is equal to the probability of keeping less than our test statistic of -2.473 and that's it. So if we have -2.743, the p value for this test is the probability of getting a test statistic even smaller than that. Now let's use Excel to calculate this again, we must note that the degrees of freedom will be 40 And we get .0089. Now moving on to see looking at the alternative hypothesis, it's that mu is not equal to a certain value. Which again means the probability of getting a test statistic at least as extreme as the one we have. So that is the probability that T. Is greater than 1.491 Plus the probability that T. is less than -1.491. And going back to Excel First note that N is equal to 30. So the degrees of freedom is 29. That's -1 491 20° of freedom. And we're looking for a cumulative probability. So we enter true and then hit enter And we get .076 and then last for d The alternative hypothesis is greater than so the p value is the probability of tea being greater than our test statistic Which was 3.635. We can re express this as 1 -1 the probability that he is less than 3.635. And going back to Excel. So the Test statistic is 3.635. The degrees of freedom is three because N was four. And again we're looking for a cumulative distribution. So that's true. .982. So this comes out to .018. Okay and by the way I just noticed An error in this one. We I just calculated this, but I forgot about this second term. So we have to multiply this p value times two, And that comes out 2.14 seven approximately. Okay. I guess that's it. I hope this works for you.
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