Question

question 2
the local power company is testing a new type of thermostat that automatically changes the air - conditioning settings on peak days to see if using the new thermostat will reduce power consumption. the power company randomly selects 35 homes of a similar size to test the new thermostat. they compare the average summer power consumption before and after the new thermostat is installed. the power company wishes to find out if there is evidence to suggest that energy use was significantly higher before the new thermostats were installed (with lower energy use after the new thermostats were installed).
given the design of the study and the question of interest, which one of the following four computer outputs is relevant to use?
a.
paired t - test
sample n mean stdev se mean
before 35 8 1.4142136 0.23904572
after 35 7.5714286 1.6140143 0.2728182
differences 35 0.42857143 0.77784447 0.13147971
t - test of mean difference = 0 (vs.>0): t - value = 3.2596, p - value = 0.0013
b.
two sample t - test
sample n mean stdev se mean
before 35 8 1.4142136 0.23904572
after 35 7.5714286 1.6140143 0.2728182
difference = mu (before)-mu (after)
estimate for difference: 0.429
t - test of difference = 0 (vs>): t - value = 1.1815 p - value = 0.1208 df = 68
c.
paired t - test
sample n mean stdev se mean
before 35 8 1.4142136 0.23904572
after 35 7.5714286 1.6140143 0.2728182
difference 35 0.42857143 0.77784447 0.13147971
t - test of mean difference = 0 (vs.≠0): t - value=-3.2596, p - value = 0.0025
d.
two sample t - test
sample n mean stdev se mean
before 35 8 1.4142136 0.23904572
after 35 7.5714286 1.6140143 0.2728182
difference = mu (before)-mu (after)
estimate for difference 0.429
t - test of difference = 0 (vs not =): t - value = 1.1815 p - value = 0.2415 df = 68

Explanation:

Step1: Identify the type of test

The power company is comparing the power - consumption of the same 35 homes before and after the installation of a new thermostat. This is a paired - samples situation as the data points are related (same homes). So, we should use a paired t - test, eliminating options B and D which are two - sample t - tests.

Step2: Determine the alternative hypothesis

The power company wants to find out if energy use was significantly higher before the new thermostats were installed. The null hypothesis $H_0:\mu_d = 0$ (where $\mu_d$ is the mean of the differences between before and after) and the alternative hypothesis $H_1:\mu_d>0$.

Step3: Check the t - value and p - value for the correct alternative

For a one - tailed paired t - test with $H_1:\mu_d>0$, we look for a positive t - value and a p - value that is appropriate for the one - tailed test. Option A has a paired t - test with a positive t - value ($T - value = 3.2596$) and a p - value of $0.0013$ for the alternative $H_1:\mu_d>0$. Option C has a paired t - test but the alternative hypothesis is two - tailed ($\mu_d
eq0$) which is not what we want.

Answer:

A. Paired T - Test
Sample N Mean StDev SE Mean
Before 35 8 1.4142136 0.23904572
After 35 7.5714286 1.6140143 0.2728182
Differences 35 0.42857143 0.77784447 0.13147971
T - test of mean difference = 0 (vs. > 0): T - value = 3.2596, p - value = 0.0013