Question

Here is the start of the SAS code to address this question:

DATA READING;
INPUT GROUP $ WORDS @@;
DATALINES;
X 700 X 850 X 820 X 640 X 920
Y 480 Y 460 Y 500 Y 570 Y 580
Z 500 Z 550 Z 480 Z 600 Z 610
;

Is there a significant difference between the mean reading score among the groups?

1. Yes
2. No

Assuming you answer "yes" in the previous question, between what groups was there a significant difference in means?

1. X and Z
2. Y and Z
3. X and Y
4. Two of the above are correct
5. All three of the above are correct

Answer 1

The means of the 3 groups are:

= 786,   = 518,   = 548

And the std deviations are:

s_x = 113.93, s_y = 54.037, s_z = 58.05

The 2-sample t-statistic for samples X and Y =

= [ - ] / sqrt[(s_x²/n1) + (s_y²/n2)]

4.752 > t_{0.975, 8} = 2.262 (critical 2-sided t-score for 5% significance and 8 degrees of freedom)

Hence, we can reject the null hypothesis that the means are equal at the 5% significance level. So, there is a significant difference between means of X and Y.

Similarly, for X and Z, t-statistic = 4.162 > t_{0.975, 8} = 2.262. Hence, there is a significant difference between means of X and Z as well.

Now, for Y and Z, t-statistic = -0.846 > t_{0.025, 8} = -2.262 (we look at the left tail probability as the t-statistic lies towards the left of the mean 0). Hence, we cannot reject the null hypothesis that there is a significant difference between means of Y and Z at the 5% significance level.

Hence

• Yes, There is a significant difference among the means reading scores of the 3 groups
• There is a significant difference in means between groups X and Y, and between X and Z, but not between Y and Z.