Question

Assume that the paired data came from a population that is normally distributed. using a 0.05 significance level and dequalsxminus​y, find d overbar​, s subscript d​, the t test​ statistic, and the critical values to test the claim that mu subscript dequals0. statcrunch

Answer 1

"Assume that the paired data came from a population that is normally distributed. Using a 0.05 significance level and d = (x - y), find
`\bar{d}`,
`s_(d)`, the t-test statistic, and the critical values to test the claim that
`\mu_(d) = 0`"

You did not attach the data, therefore I can give you the general explanation on how to find the values required and an example of a random paired data.

For the example, please refer to the attached picture.

A) Find
`\bar{d}`
You are asked to find the mean difference between the two variables, which is given by the formula:

`\bar{d} = (\sum (x - y))/(n)`

These are the steps to follow:
1) compute for each pair the difference d = (x - y)
2) sum all the differences
3) divide the sum by the number of pairs (n)

In our example:

`\bar{d} = (6)/(8) = 0.75`

B) Find
`s_(d)`
You are asked to find the standard deviation, which is given by the formula:

`s_(d) = \sqrt{ \frac{\sum(d - \bar{d}) }{n-1} }`

These are the steps to follow:
1) Subtract the mean difference from each pair's difference
2) square the differences found
3) sum the squares
4) divide by the degree of freedom DF = n - 1

In our example:

`s_(d) = \sqrt{ (101.5)/(8-1) }`
= √14.5
= 3.81

C) Find the t-test statistic.
You are asked to calculate the t-value for your statistics, which is given by the formula:

`t = \frac{(\bar{x} - \bar{y}) - \mu_(d) }{SE}`

where SE = standard error is given by the formula:

`SE = ( s_(d) )/( √(n) )`

These are the steps to follow:
1) calculate the standard error (divide the standard deviation by the number of pairs)
2) calculate the mean value of x (sum all the values of x and then divide by the number of pairs)
3) calculate the mean value of y (sum all the values of y and then divide by the number of pairs)
4) subtract the mean y value from the mean x value
5) from this difference, subtract
`\mu_(d)`
6) divide by the standard error

In our example:
SE = 3.81 / √8
= 1.346

The problem gives us
`\mu_(d) = 0`, therefore:
t = [(9.75 - 9) - 0] / 1.346
= 0.56

D) Find
`t_(\alpha / 2)`
You are asked to find what is the t-value for a 0.05 significance level.

In order to do so, you need to look at a t-table distribution for DF = 7 and A = 0.05 (see second picture attached).

We find
`t_(\alpha / 2) = 1.895`

Since our t-value is less than
`t_(\alpha / 2)` we can reject our null hypothesis!!