P(-1.97<t<1.97)

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for this question for part A We are asked to consider AT distribution with 7° of freedom. Now we want the probability probability that T is between -1.97 And 1.97. We can re express this this way, it's the probability that T Is less than 1.97 minus the probability that T His listener listener equal to -1.97. Now, since the T distribution is a continuous distribution, it doesn't really make any difference whether we have a lesson or equal to sign here or just a lesson sign. We can also note that each of these terms Are a cumulative probabilities, probability that T is less than 1.97. The probability that T is less than or equal to -1.97. These are cumulative probabilities and these can easily be solved and excel. So in an excel spreadsheet you type equals and then T dot i n v r t dot d i S T. And so for the first argument we put the value. So let's say we're going to find the probability that T is less than 1.97 X is 1.97 2nd. We put the degrees of freedom which is seven. And then for a cumulative we put true since this is a cumulative probability that we're trying to find. So that's the first part of our expression then we have minus and again we enter -1.97 This time and seven and again it's true And we get .9105. Now, just to show you the Excel function compared to what we have written down here. So the first term is the first term here. And the second term is this term here and we have a part be considered a T distribution With 13° of freedom. And we want to find C. Such that the probability that T is greater than C Is 0.10. one way to do this is to go to a tea table and we have 13° of Freedom. We want an area or probability of .1 in the upper tail. So that should actually be this column here. So it means that C. is equal to 1.350. So we found from the table the probability that t is at least 1.350 is equal to .10, So C is equal to .1 point 350. Now we could also do this in Excel. So the formula this time is equals T dot I envy. So this is the inverse um computation using the T distribution. Now you can see here it returns a left tailed inverse of the student's t distribution. We're looking for the probability that key is greater than C being equal to 0.1. So that's the probability is greater than C. So that's the right tail. So in this case, since this uses a left tailed Um computation, the probability that we want to put in is not .1 but it's 0.9 1 -1 -1 And the degrees of freedom is 13 and we hit enter and we get 1.350. So these are the answers. I guess you can around this to three decimal places. Its point 911. I hope I didn't make that too confusing about how the T distribution works in Excel versus on the table. But let's just say that this is a T. Distribution and suppose C. Is here in the table. In the at the top of the table, numbers that are shown are areas in the distribution to the right of P. So I'm talking about these numbers here So we want to see such that this area is .1. However, in an Excel it finds C. Such that the area to the left of the value is the given probability that we enter into the function. So when we used Excel we needed to enter .9 and to the probability

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