Question

For the following questions, you will use the following data on particulate matter in air samples:

Site	Sample Size	Mean	Std. Dev.
1	8	36.25	14.56
2	8	36.00	2.88

You will test for a significant difference between means at the alpha=0.05 level.

Calculate the standard error of the mean difference.

Calculate the test statistic.

What are the degrees of freedom?

Should you reject or retain the null hypothesis

In a study on differences in urinary creatinine levels between early and late stage diabetics, an independent two-sample t test yielded a p-value of 0.03. Which of the following are correct conclusions from this p-value? choose all that apply.

a. Reject the null hypothesis at alpha=0.05.

b. Reject the null hypothesis at alpha=0.01.

c. More extreme differences between groups in the study would have resulted in a p value larger than 0.03.

d. The probability of observing data as extreme as what was found in the study if the null hypothesis is true is 0.03.

Answer 1

Standard error of the mean difference is,

SE = sqrt[ (s1²/n1) + (s2²/n2) ]

where s1 is the  standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

SE = sqrt[ (14.56²/8) + (2.88²/8) ] = 5.247476

Test statistic is,

t = (x1 - x2) / SE

= (36.25 - 36.00) / 5.247476 = 0.0476

Degree of freedom is,

DF = (s1²/n1 + s2²/n2)² / { [ (s1² / n1)² / (n1 - 1) ] + [ (s2² / n2)² / (n2 - 1) ] }

= (14.56²/8 + 2.88²/8)² / { [ (14.56² / 8)² / (8 - 1) ] + [ (2.88² / 8)² / (8 - 1) ] }

= 8 (Rounded to nearest integer)

P(t > 0.0476, df = 8) = 0.4816

For two tail test, p-value = 2 * 0.4816 = 0.9632

As, p-value is greater than alpha=0.05 level, we fail to reject the null hypothesis and retain the null hypothesis.

Part 2 -

As, p-value is less than alpha=0.05 level, we reject the null hypothesis.

P-value = 0.03. This suggests that probability of observing data as extreme as what was found in the study if the null hypothesis is true is 0.03.

Hence the correct options are

a. Reject the null hypothesis at alpha=0.05.

d. The probability of observing data as extreme as what was found in the study if the null hypothesis is true is 0.03.