Question

Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does eye grease work?   In one study, 16 student subjects took a test of sensitivity to contrast after 3 hours in bright sun, both with and without eye grease. This is a repeated measure design where each subject was measured under each condition.   The differences in sensitivity (i.e. diff = with eye grease score – without eye grease score) are:

0.07     0.64    -0.12    -0.05    -0.18    0.14    -0.16    0.03

0.05     0.02     0.43     0.24     -0.11    0.28     0.05    0.29

         To be clear, positive differences means that glare is reduced, while negative differences means that glare is increased. Suppose that the subjects are a simple random sample of all young people with normal vision.

a.      State the null and alternative hypotheses in words and symbols to test whether or not eye grease reduces glare.   You should state the hypotheses in terms of mean difference. Let this be a 1-tailed test. Hint:   Remember what it means to be a Null Hypothesis – how does this translate into a number?

b.     To use StatKey, you must first enter the data.   To do this,

·       start StatKey

·       Click on Test for Single Mean under Randomization Hypothesis Tests

·       Click on EDIT DATA

·       In the first line, change the variable name to:   diff.    Be sure the box “data has header row” is checked.

·       Delete any data in the EDIT DATA template.

·       Type in the data from above, one number per line.   Please pay careful attention to negative signs and where the decimal is located.

·       Click on OK

·       Change the Null Hypothesis Mean value to the value you hypothesized in part a)

Report the sample size (n), the sample mean, and standard deviation for the Original Sample.   This information should come from StatKey.

c.      Use StatKey to set up the Randomization Distribution for this the hypothesis test in part a). Generate at least 5,000 samples. Then use the Randomization Distribution to find the p-value for this one-tailed hypothesis test. Report the p-value and interpret this hypothesis test in the context of this problem. You are to use whatever value of ?=0.05 for this test.

d.     In the Randomization Test for a Mean, click on EDIT DATA, highlight, and copy the Eye Grease Data you entered. Now, you are going to use StatKey to find a 95% Bootstrap confidence interval for the true mean difference, and you need to transfer the data over. To do this:

·       Click on the STATKEY icon in the upper left corner to go back to the beginning of StatKey.

·       Click on “CI for Single Mean…….” In the Bootstrap Confidence Interval area.

·       Click on EDIT DATA.

·       Highlight whatever data is there, then click CTRL and V at the same time. Then click on OK.   This will populate the Bootstrap CI for a “Custom Dataset.”

·       Generate at least 5,000 samples.    Just keep clicking on the generate 1000 samples option 5 times.

Report the confidence interval and discuss what it means in the context of this problem. Be sure to point out how this confidence interval relates to the Null hypothesis.   

Answer 1

(1)

Let be the mean difference in sensitivity with eye grease minus without eye grease.

The null and alternative hypotheses are,

Null hypothesis

Alternative hypothesis

Correct option is B.

is the mean sensitivity difference in the population

Correct option is C.

Part 1

Null hypothesis

Alternative hypothesis

is the mean sensitivity difference in the population

Explanation | Common mistakes | Hint for next step

Null and alternative hypothesis plays an important role in taking the decision about the claim. In this context, the null and alternative hypotheses are,

Null hypothesis

Alternative hypothesis

(2)

From the given information, the subjects are a simple random sample of all young people with normal vision, that contrast differences follows a normal distribution in this population.

The statement is true.

The objective is to test whether eye grease has a significant impact on eye sensitivity.

Part 2

The objective is to test whether eye grease has a significant impact on eye sensitivity.

Explanation | Hint for next step

The statement is true.

The objective is to test whether eye grease has a significant impact on eye sensitivity.

(3)

Calculate the mean given data as follows:

Mean:

From the given information,

Test statistic:

(4)

The p-value of the test is obtained as follows:

The right-tailed p-value is,

The p-value of the test is 0.173.

The p-value of the test is 0.173.

The p-value is greater than the level of significance 0.05. So the null hypothesis is not rejected.

Therefore, there is no evidence to conclude that the sensitivity of eye grease is greater than without eye grease.

The data do not provide good evidence that eye grease increases sensitivity.