Question

In an effort to link cold environment to an increase in mean blood pressure, two random samples of 5 rats each were exposed to different environments. One sample of rats was held in a normal environment of 26◦C and the other was held aat 15◦C.

Blood pressures were measured for rats of both groups after 1 day and are given below:

26◦C BP: 214 194 221 198 212

15◦C BP: 225 215 253 272 254

(a) If the scientists want to assume that the necessary populations are normal, what test[s] would be reasonable to run? Explain why. Identify the hypotheses of interest and then run this/these tests in R and report the test statistic, degrees of freedom, and resulting p value (it may also be useful to compute these by hand for practice, but we will not be grading you on it).

(b) If the researcher does not want to assume that the relevant populations are normally distributed, what 3 tests could they perform?

(c) Perform each of the three tests listed above (using set.seed(1) and B=10000 for any that require computer simulation) and (i) identify the assumptions (ii) report the observed test statistic and (iii) the resulting p value.

(d) Based on your findings from the tests in parts a and b, what conclusion would you draw and what recommendations would you give to the scientist?

Answer 1

Define , $\mu_1$ : Population mean blood pressure of rats exposed to 15 C environment .

μ2
: Population mean blood pressure of rats exposed to $26^{\circ}$ C environment .

  $\bar{x}_1$ :  Sample mean blood pressure of rats exposed to15 C environment .

  $\bar{x}_2$ :  Sample mean blood pressure of rats exposed to $26^{\circ}$ C environment .

  $s_1$ : Sample standard deviation of rats exposed to 15 C environment .

  $s_2$ : sample standard deviation of rats exposed to  $26^{\circ}$ C environment .

To test :

$H_0:\mu_1=\mu_2\ \ vs. \ \ H_1:\mu_1>\mu_2$

Test statistic :

$t^{*}=\frac{{\bar{x}}_1-{\bar{x}}_2}{s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\overset{H_0}{\sim}t_{n_1+n_2-2}$

where , $\bar{x}_1= 378.5 \ , \bar{x}_2= 165.8333 \ , s_1= 23.9562\ , s_2= 14.7705$ and

$s_p =\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}=19.9006$

Degrees of freedom = 10

We reject the null hypothesis or H₀ if the observed value of $t^*>t_{10}\left ( \alpha \right )$ , where , $\alpha=0.05$

The value of the test statistic : $t^*=18.509$

Critical value : $t_{10}\left ( \alpha \right )=1.812$

Now , $t^*=18.509>1.812$

Hence we reject the null hypothesis or H₀ .

Thus we have sufficient evidence to support the claim that rats exposed to a 15°C environment have a higher mean blood pressure than rats exposed to a 26°C environment .