Question

If t follows a t7 distribution, find t0 such that (a) p(|t | < t0) = .9 and (b) p(t > t0) = .05.

Answer 1

Answer:

a) $P(-t_o < t_7$

Using the symmetrical property we can write this like this:

$1-2P(t_7$

We can solve for the probability like this:

$2P(t_7$

$P(t_7$

And we can find the value using the following excel code: "=T.INV(0.05,7)"

So on this case the answer would be $t_o =\pm 1.895$

b) For this case we can use the complement rule and we got:

$1-P(t_7$

We can solve for the probability and we got:

$P(t_7$

And we can use the following excel code to find the value"=T.INV(0.95;7)"

And the answer would be $t_o = 1.895$

Step-by-step explanation:

Previous concepts

The t distribution (Student’s t-distribution) is a "probability distribution that is used to estimate population parameters when the sample size is small (n<30) or when the population variance is unknown".

The shape of the t distribution is determined by its degrees of freedom and when the degrees of freedom increase the t distirbution becomes a normal distribution approximately.  

The degrees of freedom represent "the number of independent observations in a set of data. For example if we estimate a mean score from a single sample, the number of independent observations would be equal to the sample size minus one."

Solution to the problem

For this case we know that $t \sim t(df=7)$

And we want to find:

Part a

$P(|t|$

We can rewrite the expression using properties for the absolute value like this:

$P(-t_o < t_7$

Using the symmetrical property we can write this like this:

$1-2P(t_7$

We can solve for the probability like this:

$2P(t_7$

$P(t_7$

And we can find the value using the following excel code: "=T.INV(0.05,7)"

So on this case the answer would be $t_o =\pm 1.895$

Part b

$P(t>t_o) =0.05$

For this case we can use the complement rule and we got:

$1-P(t_7$

We can solve for the probability and we got:

$P(t_7$

And we can use the following excel code to find the value"=T.INV(0.95;7)"

And the answer would be $t_o = 1.895$