Find the following probabilities for a sample size of 75:(A) P(T<−1.65)=P\left(T<-1.65\right)= _____________(B) P(T>1.65)=P\left(T>1.65\right)= _____________(C) P(T<−1.65P\left(T<-1.65\right. OR T>1.65)=T>1.65)=_{} ____________

(A) 0.0516 0.0516 0.0516 (B) 0.0516 0.0516 0.0516 (C) 0.1032 0.1032 0.1032

Find the following probabilities for a sample size of 75: (A) P ( T < − 1.65 ) = P\left(T<-1.65\right)= _____________ (B) P ( T > 1.65 ) = P\left(T>1.65\right)= _____________ (C) P ( T < − 1.65 P\left(T<-1.65\right. OR T > 1.65 ) = T>1.65)=_{} ____________

Welcome to the practice. In this problem, we'd like to find the following probabilities for a sample size of 75, starting with in part a, the probability of getting a t score below negative 1.65. Because I wanna find the probability of getting a t score below a certain value, I have this less than sign here. This is a left tail probability, so I wanna use equals t dot dist to find it. I can either copy and paste in the function where I want the probability to appear or I can type it in equals t dot dist open parenthesis. Then I'm ready to add in my inputs starting with the first one, which is the t score from the problem. My t score is negative 1.65, so I'll put that in next. I'll do a comma, and then I wanna put in the degrees of freedom, which is the sample size minus one. I can see that the sample size is 75, so I can type in 75 minus one. Or I know that that's 74, so I could also type in 74. Then I'll do a comma, and I know the last input is true. So I'll type in true for this final input. I can click enter, and we end up getting the probability at about 0.05. For part b, I'd like to find the probability of getting a t score above 1.65. Because I have a greater than sign here and I wanna find the probability of getting above a certain t score, I can use the equals t dot dist dot r t function. So I'm gonna come over to where I want it, and I'm gonna type it in equals t dot dist dot r t. But I could, of course, also copy and paste in my function. I'll put that open parenthesis, and then I'll start putting in my inputs. I have two this time starting with the t score, which I can see in the problem is 1.65, and then we'll put in the degrees of freedom, which again is 75, the sample size, minus one. I can click enter, and we get the probability at about 0.05. Next, in part c, I wanna find the probability of getting a t score below negative 1.65 or above 1.65. So we can see that this is going to be a two tailed probability. We actually already found the probability of each of our two tails in parts a and b. So I could add them together, and I would expect that this answer would be about point one because that would be about the sum of these two probabilities, the sum of the probability in each tail. But I'm actually going to find it using this function here. So I'm gonna come over to where I want my probability and either copy and paste in the function or type it in. Equals t dot dist dot two t open parenthesis. Now I'll put in the t score, and as a reminder, you need to put in a positive one, so 1.65. I'll do a comma, and then I need to add my degrees of freedom, which is the sample size, 75 minus one. Clicking enter will get us that probability at about 0.1, which is exactly what we expected it to be. Great work.

Find the following probabilities for a sample size of 75: (A) P ( T < − 1.65 ) = P\left(T<-1.65\right)= _____________ (B) P ( T > 1.65 ) = P\left(T>1.65\right)= _____________ (C) P ( T < − 1.65 P\left(T<-1.65\right. OR T > 1.65 ) = T>1.65)=_{} ____________

(A) 0.0516 0.0516 0.0516 (B) 0.0516 0.0516 0.0516 (C) 0.1032 0.1032 0.1032

(A) 0.9484 0.9484 0.9484 (B) 0.9484 0.9484 0.9484 (C) 0.0516 0.0516 0.0516

(A) 0.0516 0.0516 0.0516 (B) 0.0516 0.0516 0.0516 (C) 0.0516 0.0516 0.0516

(A) 0.9484 0.9484 0.9484 (B) 0.9484 0.9484 0.9484 (C) 0.8968 0.8968 0.8968

7. Sampling Distributions & Confidence Intervals: Mean

Finding Probabilities and T Critical Values - Excel

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Finding Probabilities and T Critical Values - Excel

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Multiple Choice

Find the following probabilities for a sample size of 75:
(A) $P (T < - 1.65) =$ _______
(B) $P of open paren T is greater than 1.65 close paren equals$ _
(C) $P open paren T is less than negative 1.65$ OR $T is greater than 1.65 close paren equals sub$ ______

(A) $0.0516$ (B) $0.0516$ (C) $0.1032$

(A) $0.9484$ (B) $0.9484$ (C) $0.0516$

(A) $0.0516$ (B) $0.0516$ (C) $0.0516$

(A) $0.9484$ (B) $0.9484$ (C) $0.8968$

Verified step by step guidance

Recognize that the variable T follows a t-distribution with degrees of freedom equal to the sample size minus one, so $ df = 75 - 1 = 74 $.

For part (A), to find $ P(T < -1.65) $, use the t-distribution table or a calculator to find the cumulative probability to the left of $ -1.65 $ with 74 degrees of freedom.

For part (B), to find $ P(T > 1.65) $, note that this is the complement of the cumulative probability up to 1.65, so calculate $ 1 - P(T \leq 1.65) $ using the t-distribution with 74 degrees of freedom.

For part (C), to find $ P(T < -1.65 \text{ or } T > 1.65) $, recognize this is the sum of the two tail probabilities from parts (A) and (B), so add $ P(T < -1.65) + P(T > 1.65) $.

Use symmetry properties of the t-distribution if needed, since the t-distribution is symmetric about zero, which means $ P(T < -1.65) = P(T > 1.65) $, simplifying calculations.

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