Question

The following scores resulted on a memory test that was administered to five subjects in two different conditions (A and B) to check if there is a difference between the two conditions: Subject_ID A B

1 20 15

2 10 4

3 4 4

4 11 8

5 15 9

a) Find the estimated mean difference. b) Find the sample standard deviation of the differences. c) Find the standard error of the sample mean difference between conditions A and B. d) Determine a 95% confidence interval for the mean difference. e) Find the observed test statistic. f) What is the p-value for this test statistic? g) Assuming α = .05, what is your reject region? h) Is there a statistically significant difference between the means at α = .05? State your conclusion.

Answer 1

Solution: a) Find the estimated mean difference.

Answer:

b) Find the sample standard deviation of the differences.

Answer: The sample standard deviation of the differences is given below:

$s=\sqrt{\frac{(5-4)^2+(6-4)^2+(0-4)^2+(3-4)^2+(6-4)^2}{5-1}}=\sqrt{\frac{1+4+16+1+4}{4}}=2.55$

c) Find the standard error of the sample mean difference between conditions A and B.

Answer: The standard error of the sample mean difference between conditions A and B is:

$S.E=\frac{s}{\sqrt{n}}=\frac{2.55}{\sqrt{5}}=1.14$

d) Determine a 95% confidence interval for the mean difference.

Answer: The 95% confidence interval is:

$\bar{d}\pm t_{\frac{0.05}{2}} \times S.E$

$4\pm 2.776 \times 1.14$

$4\pm 3.16$

$\left [ 4- 3.16,4+ 3.16 \right ]$

$\left [ 0.84,7.16 \right ]$

Therefore, the 95% confidence interval is $\left [ 0.84,7.16 \right ]$

e) Find the observed test statistic.

Answer: The observed test statistic is:

$t=\frac{\bar{d}}{S.E}=\frac{4}{1.14}=3.51$

f) What is the p-value for this test statistic?

Answer: The p-value using the t distribution is given below:

$p-value=0.0247$

g) Assuming α = .05, what is your reject region?

Answer: The rejection region is the area less than $t=-2.776$ and the area more than $t=2.776$

h) Is there a statistically significant difference between the means at α = .05? State your conclusion.

Answer: Since the p-value is less than the significance level 0.05, therefore, we reject the null hypothesis and conclude that there is a difference between the two conditions.