Question

Listed below are systolic blood pressure measurements​ (mm Hg) taken from the right and left arms of the same woman. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Use a 0.10 significance level to test for a difference between the measurements from the two arms. What can be​ concluded? Right arm 152 138 135 139 132 Left arm 178 176 176 142 151

Answer 1

There is a significant difference between the two means based on this samples at the 0.10 level of significance.

Explanation:

Let's call

`\bf \mu_r` mean of the systolic pressure from the right hand

`\bf \mu_l` mean of the systolic pressure from the left hand

and construct a confidence interval for the difference

`\bf \mu_r - \mu_l`

based on the sample of size 5.

The confidence interval whose endpoints are

`\bf (\bar x_r -\bar x_l)\pm t^*\sqrt{(s_r^2)/(5)+(s_l^2)/(5)}`

where

`\bf \bar x_r` = mean of the sample from the right hand

`\bf \bar x_l` = mean of the sample from the left hand

`\bf s_r` = standard deviation of the sample from the right hand

`\bf s_l` = standard deviation of the sample from the left hand

`\bf t^*` = t-score corresponding to a level of significance 0.10 or a confidence level 90%

Since the sample is too small we have better use the Student's t-distribution with 4 (sample size -1) degrees of freedom, which is the approximation of the Normal distribution for small samples.

For a 90% confidence level
`\bf t^*` equals 2.132

Let's compute now the means and standard deviations of the samples

From the right hand we have

`\bf \bar x_r` = 139.2

`\bf \s_r` = 7.66

From the left hand we have

`\bf \bar x_l` = 164.6

`\bf \s_l` = 16.85

Then our confidence interval would be

`\bf (\bar x_r -\bar x_l)\pm z^*\sqrt{(s_r^2)/(5)+(s_l^2)/(5)}=-25.4\pm 2.132*8.28`

finally, the interval is

[-43.05, -7.75]

Since our confidence interval does not contain the zero, we can say there is a significant difference between the two means based on this samples at the 0.10 level of significance.