Question

A scientist wishes to investigate whether exposure to sunlight reduces the amount of time it takes for a particular chemical reaction to take place. There is natural variability in reaction time. Data are recorded for 20 different experiments, 10 instances of reaction time in bright, and 10 instances of reaction time in shade. These are presented in the following table.

Experiment	Conditions	Time	Experiment	Conditions	Time
1	Bright	7.1	11	Shade	7.4
2	Bright	6.2	12	Shade	7.0
3	Bright	8.1	13	Shade	8.1
4	Bright	7.4	14	Shade	8.9
5	Bright	7.2	15	Shade	7.1
6	Bright	6.4	16	Shade	7.0
7	Bright	6.5	17	Shade	7.5
8	Bright	6.7	18	Shade	8.6
9	Bright	6.8	19	Shade	7.3
10	Bright	8.0	20	Sade	6.9

The scientist does a statistical test and obtains the output in the following display.

Two sample T-test and Confidence Interval

Two sample T for Bright vs Shade

	N	Mean	St Dev	SE Mean
Bright	10	7.040	0.648	0.21
Shade	10	7.580	0.710	0.22

95% CI for mu Bright – mu Shade: (-1.18,    0.10)

T-Test mu Bright = mu Shade (vs not =): T = -1.78      P = 0.093     DF=18

Both use Pooled StDev = 0.680

The scientist really believes that reactions in bright are on average quicker. But according to the analysis results, it will not be possible to publish anything. The scientist then goes to see a statistician to find out if there is anything that can be done to change the conclusion. The statistician flippantly suggest that a o ne-sided test would do the trick, but emphasises that a better solution by far would involve collecting more data.

1. Explain why a one-sided test will change the conclusion, if a predetermined significance level of 0.05 is chosen

2. Explain the two types of error that may be made when testing. How does the sample size affect the probability of making each one, if a predetermined significance level of 0.05 is chosen?

3. The scientist wants to detect the difference of +/- 0.3 seconds between the mean reaction times. Set the significance level to be 0.05 and the power to be 0.95. It is agreed that more experiments will be run. Do a sample size calculation to determine how many more experiments of each type should be run

Answer 1

1)

The t-sided test is used when we do not know the direction of relationship between two means such that whether the difference in means is significantly greater than or less than zero at predetermined significance level. For example if we are using 5% significance level then the two-sided alternative hypothesis will test the claim of difference in means at 2.5% significance level at both sides.

But if we know that the direction of the relationship between two means we can use the one-sided alternative hypothesis(greater than or less than). In this case, if the predetermined significance level is 5% then the one-sided alternative hypothesis will test the claim of difference in means at 5% significance level.

This is the reason why one-sided test will change the conclusion.

2)

Before explaining the two types of error, the Statistical significance defines that the obtained results are not due to change such that there are some statistical evidence of getting that result.

Type I error -

“A Type I error is the probability of rejecting the null hypothesis when the null hypothesis is actually TRUE”. It is the probability of correctly rejecting the null hypothesis. Type I error can be defined by significance level . The can be defined as a 5% significance level which states that there is 5% chance of making type I error or there is 5% chance of rejecting the null hypothesis when it is true.

For larger sample size, we can reject/not reject the null hypothesis more effectively such that as the sample size increase, the difference in mean can be define more accurately hence improves the type I error (P-value of the test).

Type II error -

“A Type II error is the probability of failing to reject the null hypothesis when the null hypothesis is actually FALSE. Type II error is denoted by . While the value can be defined as probability of rejecting the null hypothesis when the null hypothesis is False also called power of the test.

The sample size affects the type II error such that as the sample increases type II decreases. We can say that as the sample size increase, the probability of failing to reject the null hypothesis when it is true will decrease.

3)

Yes, the sample size calculation should be run to get the desired power of the test which mean to correctly reject the false null hypothesis, the sufficient sample must be need.