Question

A question of medical importance is whether jogging leads to a reduction in one’s pulse rate. To test this hypothesis, 8 nonjogging volunteers agreed to begin a 1-month jogging program. After the month their pulse rates were determined and compared with their earlier values. If the data are as follows, can we conclude that jogging has had an effect on the pulse rates? Subject 1 2 3 4 5 6 7 8 Pulse Rate Before 74 86 84 79 70 78 79 70 Pulse Rate After 70 85 90 110 71 80 69 74Can you conclude that jogging leads to a reduction in one's pulse rate?

Answer 1

Answer:

So the p value is higher than any significance level given, so then we can conclude that we FAIL to reject the null hypothesis that the difference mean between after and before is higher or equal than 0. So we can't conclude that jogging leads to a reduction in one's pulse rate.

Step-by-step explanation:

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 before , y = test value after

x: 74 86 84 79 70 78 79 70

y: 70 85 90 110 71 80 69 74

The system of hypothesis for this case are:

Null hypothesis: $\mu_y- \mu_x \geq 0$

Alternative hypothesis: $\mu_y -\mu_x$

The first step is calculate the difference $d_i=y_i-x_i$ and we obtain this:

d: -4, -1, 6, 31, 1, 2, -10, 4

The second step is calculate the mean difference

$\bar d= \frac{\sum_{i=1}^n d_i}{n}= \frac{29}{8}=3.625$

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

$s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1} =12.130$

The 4 step is calculate the statistic given by :

$t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{3.625 -0}{\frac{12.130}{\sqrt{8}}}=0.845$

The next step is calculate the degrees of freedom given by:

$df=n-1=8-1=7$

Now we can calculate the p value, since we have a left tailed test the p value is given by:

$p_v =P(t_{(7)}$

So the p value is higher than any significance level given, so then we can conclude that we FAIL to reject the null hypothesis that the difference mean between after and before is higher or equal than 0. So we can't conclude that jogging leads to a reduction in one's pulse rate.