Question

Assume the blood pressures of 10 people were measured before and after sleeping for the night. What is the corresponding critical t-value for a one-tailed, paired-samples t test with an alpha level of 0.05 (5%)?
Please report your answer as an absolute value and include three decimal places.

Answer 1

Answer:

For the critical value we need to calculate the degrees of freedom given by:

$df = n-1= 10-1=9$

And since we have a one tailed test we need to look in the t distribution with 9 degrees of freedom a quantile who accumulates 0.05 of the area on a tail and we got:

$|t_{crit}| =1.833$

Step-by-step explanation:

Previous concepts

A paired t-test is used to compare two population means where you have two samples in which observations in one sample can be paired with observations in the other sample. For example if we have Before-and-after observations (This problem) we can use it.  

Let put some notation  

x=test value with right arm , y = test value with left arm  

The system of hypothesis for this case are:  

Null hypothesis: $\mu_y- \mu_x = 0$  

Alternative hypothesis: $\mu_y -\mu_x \neq 0$  

The first step is calculate the difference $d_i=y_i-x_i$

The second step is calculate the mean difference  

$\bar d= \frac{\sum_{i=1}^n d_i}{n}= \frac{361}{5}$  

The third step would be calculate the standard deviation for the differences, and we got:  

$s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1}$  

The 4 step is calculate the statistic given by :  

$t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}$  

For the critical value we need to calculate the degrees of freedom given by:

$df = n-1= 10-1=9$

And since we have a one tailed test we need to look in the t distribution with 9 degrees of freedom a quantile who accumulates 0.05 of the area on a tail and we got:

$|t_{crit}| =1.833$