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

Answer:

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

Step-by-step 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

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

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

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

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.