Immediately below is the frequency of the final algebra grades received by the 26 students who studied according to the metacognitive method.  It appears to have higher frequencies in its middle, and lower frequencies on its ends - characteristics derived from those which bring about a normal distribution.

The Frequency Diagram for Final Algebra Grade
for 26 Students Who Underwent
the Metacognitive Method

To facilitate Part II of the final exam, I also included below the possible variables with which I could operationalize the additional researcher's interests regarding the 26 students who underwent the metacognitive method:

 

Metacognitive data sets and report on missing data:
For metacognitive method, 26 students were sampled and none were missing. 

Report on the means:
Means on all variables were positive numbers. 

Report on the standard deviations and variances:
The column "Std. Statistic" refers to standard deviation of that statistic, and if squared, results in the numbers found in the column labeled "Variance Statistic." 

Report on skewness:
ITED 8th Grade Math Scores and Final Algebra Grade showed negative skewness, -.424 and -.547 respectively with the latter indicating a greater degree of negativity and skewness than the former.  (Zero represents no skewness.)  All other variables showed positive skewness and those with higher values indicated higher degrees of positive skewness.

Report on kurtosis:
Those variables with a negative value of kurtosis showing a flatter or platykurtic nature included:
Middle School GPA;
Pre-instruction Math Attitude;
ITED 8th Grade Math Score; and
Final Algebra Grade.

Zero represents the mesokurtic  distribution;

Those variables with a positive value of kurtosis showing a steeper of leptokurtic nature included:
Pre-instruction Math Problem Solving; and
Pre-instruction General Problem Solving.

The descriptives show that among 23 students using lecture methods while taking algebra achieved:
a mean or average score of 2.5652 with a standard error of .20039.
The 95% confidence interval for the mean had a lower boundary of 2.1496 and an upper boundary of 2.9808.;
The median was 2.67.
The variance was .924.
The standard deviation was .96104.
The range of scores was larger, ranging from .00 to 4.00, and there were no reported scores at the 0.50 grade point average; so this group of students had an "outlier" who did not do very well on his/her final grade.
The interquartile range was 1.33.
The skewness was -1.139.
The kurtosis was 1.103.
 

The mean Final Algebra Grade for the 23 students using the lecture method was 2.5652 which was less than the mean reported earilier for those taking the lecture method of instruction.

The Frequency Diagram for Final Algebra Grade
for 23 Students Who Underwent
the Lecture Method

Lecture Method data sets and report on missing data:
For Lecture method, 23 students were sampled and none were missing. 

Report on the means:
Means on all variables were positive numbers. 

Report on the standard deviations and variances:
The column "Std. Statistic" refers to standard deviation of that statistic, and if squared, results in the numbers found in the column labeled "Variance Statistic." 

Report on skewness:
ITED 8th Grade Math Scores and Final Algebra Grade showed negative skewness, -.295 and -1.139 respectively with the latter indicating a greater degree of negativity and skewness than the former.  (Zero represents no skewness.)  All other variables showed positive skewness and those with higher values indicated higher degrees of positive skewness.

Report on kurtosis:
Those variables with a negative value of kurtosis showing a flatter or platykurtic nature included:
Middle School GPA;
Pre-instruction Math Attitude; and
Pre-instruction Math Problem Solving.
Zero represents the mesokurtic  distribution;

Those variables with a positive value of kurtosis showing a steeper of leptokurtic nature included:
Pre-instruction General Problem Solving; and
Final Algebra Grade.
 

Step 2 - Select and conduct the appropriate analysis 

After a literature review a researcher wishes to determine whether the metacognitive method of math instruction brings about higher grades in algebra than grades achieved by students subject to the lecture method of math instruction.

The researcher was provided two samples of algebra final grades.  One sample had 23 students and their performance measured by final algebra grade after having received the traditional lectures in class.  The second sample at 26 students and their performance measured by final algebra grade received after they had studied algebra according to the metacognitive approach.  The researcher has the means and standard deviations of these two samples and believes them to be random, representative samples.

Here are the means, standard deviations, and sample sizes:

Considerations:

1.  The sample sizes are less than 30, so we should not use a z-statistic; rather we should perform a t-statistic comparison of the means of the two samples

2.  We will use an alpha, or probability of Type I error of .05.

3.  One of the things we will have to know is whether or not the variances of the two groups are equal.  And it will help to have a table containing the following statistics for lecture and for metacognitive methods:
   - the number of students in each method;
   - their means;
   - their standard deviations; and
   - the standard error of the means

The Solution:


The Group Statistics Box below computed for each sample: number of students, mean, standard deviation, and standard error of mean.
(I have reproduced the table in case you cannot see the computed table.)
 

4.  State the hypotheses:

We state the null hypothesis as a one-tail test that:

H-sub-o : mu-sub-1 is less than or equal to mu-sub-2  .... or .... HO: u1<= u2

The alternative hypothesis is that :
H-sub-1: mu-sub-1 is greater than mu-sub-2        ... or ... u1 > u2

If we can reject the null hypothesis, this will enable us to claim that the metacognitive method produces statistically better results than the traditional lecture method insofar as final grade achievement is concerned.  By structuring the hypotheses in this way, these differences could be claimed as exceeding the differences that could have resulted from pure chance alone.

The Results Coach tells us:
A low significance value for the t test (typically less than 0.05) indicate that there is a significant difference between the two group means.
(We did not find this.)
If the confidence interval for the mean difference does not contain zero, this also indicates that the difference is significant.  (We did find this.)


Should we assume variances are equal or unequal?
The answer is found In the following Independent Samples Test Table, in which we find F value of .389 for Levene's test for the equality of variances, which is high and greater than 0.05, allowing us to use the row in the table called "Equal Variances Assumed."

5.  Find the critical value:

From the table above, the critical value for the t-statistic is -1.443.

6.  Compute the test value:

The computed test statistic is one-half the two-tailed value of .156 which is 0. 078.

7.  Make the decision:

Because 0.078 does not lie in the rejection region, (or because 0.078 > -1.443, we do not have enough evidence to support the claim that metacognitive training improves results over traditional lectures insofar as final grades in algebra is concerned. 

We do not have enough evidence to support the claim that metacognitive training improves results over traditional lectures insofar as final grades in algebra is concerned.

Step 3 - Interpret your results

A one-tail test carefully constructed so as, if ruled out, would rule in the researcher's hypothesis was constructed using alpha for type I error possibility of .05 was performed.  Equal variances were assumed based upon the F-value of the Levene Test.  Hence the top row of actual data beside "final algebra grade" was used.  It was necessary to "halve" the Sig. (2-tailed) value of .156 so as to perform a one-tail test.  The resulting .078 value did not, however, lie to the left on the number line of the t-value of -1.443 in a zone of rejection.

Effect size:

Recently, researchers have come to ask not only "Is the difference between sample means unlikely to have occurred by chance?", but also, "Is this difference large enough to be meaningful?"

The usual two group measure of effect size called "d" is given on page 334 of the text as equation13.14a.
I tried to use equation 13.6b on page 326 to calculate the pooled variance estimate because I knew both sample sizes and variances.  My pooled variance estimate equation 13.6b was thus [(26)(.59874) + (23)(.924)] / (26 23+2).
From the result,  .7834, I took the square root and put 0.8851 into the denominator of of equation 13.14a on text page 334.  The numerator in that equation was simply the difference in the sample means, or 2.9231 - 2.5632 = 0.3579.
Dividing .3579 by .8851 equaled .4044.  I interpreted this as meaning that being in the metacognitive group or in the traditional lecture group explained made a difference of less than one-half standard deviation, or more precisely, .4044 standard deviations.

Power:

Power was effectively established prior to collecting the data by establishing group sizes of 26 and 23 respectively.

Statistical Significance:

No statistical significance was found between the means of the two groups at alpha = .05.

Degrees of Freedom:

There were 47 degrees of freedom.  Each group having used up one degree of freedom, degrees of freedom for group sizes of 23 and 26 became 22 + 25 or 47.

Analysis Used:

The t-test was used, but failed to produce a result in the rejection region.
 

Introduction to Case #2

Here is the case after I changed it (operationalized it) to exclude Middle School GPA and
to include Middle School Math GPA  :
 

Here is the case after I changed it (operationalized it) to exclude Middle School GPA and
to include Pre-instruction Math Problem Solving  :

Pre-instruction Math Problem Solving is likely to do a better job of mirroring the additional researcher's intentions than did Middle School GPA  because he wanted experience with previous math, not a generalized previous experience as implied by Middle School GPA.

The Descriptive Statistics Box above lists the Mean, Standard Deviation, and Number of students for the dependent variable Final Algebra Grade, and for the independent variables: Method of Math Instruction, Pre-instruction Math Problem Solving, and for Pre-instruction Math Attitude.