We discussed GM testing a brake light. Suppose that this is very expensive, time-consuming and difficult and we have time and money for just 12 runs (tests) with the two lights total. There are two ways to run the experiment. The first is to obtain a random sample of 12 drivers, divide them into two groups at random and assign one group to the standard light and the second group to the new light. Suppose the table below represents the data from this
Standard light group reaction time New light group reaction time
1.00 0.92
0.33 0.27
1.25 1.10
0.56 0.49
0.90 0.74
1.30 1.18
Note that the mean reaction time for the standard light group was 0.89 seconds with a standard deviation of 0.383 seconds, and the mean reaction time for the new light group was 0.783 seconds with a standard deviation of 0.354 seconds. (Try to use Excel to verify these values.)
1) Find a 95% confidence interval for the true mean difference in reaction time (standard –new).
2) Using your interval, do you have statistical evidence that the new light differs from the old in terms of mean reaction time?
3) Suppose we have decided that a practically significant (important) difference in reaction time is 0.05 seconds (in either direction). Explain what the confidence interval tells us about the importance of any difference between these two lights.
4) Based on the data here, and this experimental setup, what can you recommend about which light to use?
A second experimental setup is to take 6 drivers at random from the population of all drivers, and test each of them on both lights. In this setup each person serves as their own “control”. The table below lists the data and the difference in reaction time between each light for each person;
Time for standard light Time for new light Difference in times
1.00 0.92 0.08
0.33 0.27 0.06
1.25 1.10 0.15
0.56 0.49 0.07
0.90 0.74 0.16
1.30 1.18 0.12
(You might notice that the times I picked for them here are the same as the table above. This is to illustrate a point!) Note that the mean difference was 0.107 seconds with a standard deviation of 0.043 seconds.
1) Explain what “each person serves as their own control” means. Why might this be a good idea?
2) Find a 95 % confidence interval for the true mean difference in reaction times between the two lights (standard –new).
3) Can we say there is statistical evidence for a difference between the two lights?
4) Remember that a practically significant difference is 0.05 seconds. Can we say that there is a practically significant difference?
5) The second design is a paired design. Explain why this type of design worked particularly well here.
6) Explain when the paired design will not work well.
HW:
Question 1
Suppose we are interested in getting an interval estimate for the proportion of people who approve of the performance of Pres. Obama. We ask a random sample of 1000 US adults and 592 say they approve.
What is the margin of error for this survey, assuming 95% confidence? (Write your answer in percent form to 1 decimal places, so 0.01=1.0 in percent terms.)
Question 2
Assume the data observed in question 1. Write down the upper limit of the 95% CI for p. Write the answer as a percentage without the % symbol, that is 53, rather than 53% or 0.53.
Question 3
Assume the data observed in question 1. Write down the lower limit of the 95% CI for p. Write the answer as a percentage without the % symbol, that is 53, rather than 53% or 0.53.
Question 4
Suppose we test to see if the proportion of people who approve of Mr Obama is different from 50%. So our null hypothesis will be that the proportion is 50% and the alternative that it is not. What conclusion can we draw?
a) We reject the null hypothesis and conclude that Mr Obama's support is less than 50%.
b) We fail to reject the null hypothesis, and conclude Mr Obama's support is greater than 50%.
c) We fail to reject the null hypothesis and conclude that it is possible Mr Obama's support is 50%.
d) We reject the null hypothesis and conclude Mr Obama's support is greater than 50%.