11.4: Practice Dependent t-test on Changes in Mindset

Let's try this full process, starting with a table of raw scores, and moving through the steps and the formula to analyze paired Mindset Quiz scores. Check out the section on growth mindset to refresh yourself on what that idea means.

Scenario

Dr. MO's student research group conducted a study in which one English faculty member tried to use journaling to improve growth mindset in a class that was one level below college-level English (so, remedial or basic skills English) at a community college. The Mindset Quiz was completed by students at the beginning of the semester, and then again at the end of the semester. We had complete data (both pretest and posttest scores) for 10 students (N=10).

Step 1: Hypotheses

Without knowing the means yet, we should just hope that the journalling intervention improved mindset. With this assumption, develop the hypotheses.

Exercise \(\PageIndex{2}\)

Develop the research hypothesis, in words and symbols. Then, determine the null hypothesis in words and symbols.

Answer

• Research Hypothesis: The average score will be higher on the Mindset Quiz at the end of the semester (Time 2, posttest) than the average score at the beginning of the semester (Time 1, pretest).
  • Symbols: \( \bar{X}_{\text {T2}} > \bar{X}_{\text {T1}} \)
• Null Hypothesis: The average score will be similar to the Mindset Quiz at the end of the semester (Time 2, posttest) than the average score at the beginning of the semester (Time 1, pretest).
  • Symbols: \( \bar{X}_{\text {T2}} = \bar{X}_{\text {T1}} \)

In this case, we are hoping that the changes we made will improve mindset scores, so we will use a one-directional research hypothesis.

Step 2: Collect Data

You may have noticed that we have yet to actually collect data for this step of the Null Hypothesis Significance Testing process, and that hasn’t changed with this scenario! However, you may collect data in your class (or your future career) in this step. For now, we’ll again identify the sample, population, and variables.

Exercise \(\PageIndex{1}\)

Answer the following questions to understand the variables and groups that we are working with.

1. Who is the sample?
2. Who do might be the population?
3. What is the IV (groups being compared)?
4. What is the DV (quantitative variable being measured)?

Answer

1. The sample is the group of 10 students who completed the Mindset Quiz at the beginning and the end of the semester (twice).
2. The population might be all community college students who haven't passed college-level English, although the specific population that you think this sample represents might be slightly different.
3. The IV is before and after the survey, the time periods. This can also be said as the pretest and post-test.
4. The DV is the Mindset Quiz.

Our critical values will once again be based on our level of significance, which we know is \(α\) = 0.05, the directionality of our test, which is one-tailed, and our degrees of freedom.

Exercise \(\PageIndex{3}\)

For our dependent-samples \(t\)-test, what is the critical t-score from the the same t-table from when we learned about one-sample t-tests (or going to the Common Critical Values Table page)?

Answer

For our dependent-samples \(t\)-test, the degrees of freedom are still given as \(df = N – 1\) in which N is the number of pairs.

Degrees of Freedom \( = N-1 = 10 - 1 = 9 \)

With a Degrees of Freedom of 9, the critical t-score is 1.833 from the critical t-score table.

Step 3: Analyze Data

You can use the information in Table \(\PageIndex{1}\) to calculate the mean and standard deviation for the pretest and posttest. To complete the dependent t-test equation, additional columns will be needed to find and sum the difference, and to square and sum the square of each difference.

Table \(\PageIndex{1}\)- Pretest and Posttest Mindset Quiz Scores

Participant	Mindset Quiz Pretest	Mindset Quiz Posttest
A	38	33
B	43	39
C	35	32
D	36	37
E	47	51
F	39	45
G	49	55
H	47	54
I	31	39
J	42	52
\(\Sigma\)	\(\Sigma\) = 407	\(\Sigma\) = 437

Although not required right now, you might be interested in the actual means for the Mindset Quiz at the pretest and posttest, and now you can calculate them!

For the pretest:

\[ \dfrac{\Sigma{X}}{N} = \dfrac{407}{10} = 40.70 \nonumber \]

This score means that the students had many growth mindset ideas, but still held some fixed beliefs.

For the posted:

\[ \dfrac{\Sigma{X}}{N} = \dfrac{437}{10} = 43.70 \nonumber \]

This score also shows that the students had many growth mindset ideas, but still held some fixed beliefs. However, scores of 45 or higher are considered having only growth mindset beliefs, so the posttest group was close!

Exercise \(\PageIndex{4}\)

Use the dependent t-test formula to calculate a dependent t-test.

Answer

To use the dependent t-test formula, you will need to calculate the Difference scores between each participant's pretest and posttest, find how different those Difference scores are from the mean of the difference, and square how different the Difference scores are from the mean of the difference. Table \(\PageIndex{2}\) is provided to help complete this step for each participant. The sum for each column is provided so that you can check your subtraction, squaring, and adding, but try to do it yourself so that you know how!

Table \(\PageIndex{2}\)- Mindset Quiz Pretest and Posttest Scores with Columns for Calculating

Participant	Pretest	Posttest	Difference (Posttest minus Pretest)	Difference minus Mean of Difference	Square of the Difference minus Mean of Difference
A	38	33
B	43	39
C	35	32
D	36	37
E	47	51
F	39	45
G	49	55
H	47	54
I	31	39
J	42	52
\(\Sigma\)	\(\Sigma\) = 407	\(\Sigma\) = 437	\(\Sigma\) = 30	\(\Sigma\) = 0.00	\(\Sigma\) = 262.00

Based on this table, we have what we need to complete the dependent t-test formula:

\[t = \cfrac{\left(\cfrac{\Sigma X_{D}}{N}\right) \times \left(\sqrt{N}\right)} { {\sqrt{\left(\cfrac{\sum\left((X_{D}-\overline{X}_{D})^{2}\right)}{(N-1)}\right)} }} \nonumber \]

So we fill in all of the numbers. Notice that the all of the sums were computed on the Table \(\PageIndex{2}\). Dr. MO even did the sums for you, although you should practice filling in the table and doing all of the adding so that you can do it on your own when needed. Here, we are only plugging in the answers from the table into the equation.

\[ t_{numbers} = \cfrac{ \left(\cfrac{30}{10 }\right)\left(\sqrt{10 }\right)} { {\sqrt{\left(\cfrac{262}{(10-1)}\right)} } } \nonumber \]

Then we start calculating each set of parentheses:

\[ t_{Parentheses } = \cfrac{\left(3 \times 3.16\right)} { {\sqrt{\left(\cfrac{262}{9}\right)} } } \nonumber \]

And a few more parentheses:

\[ t_{ParenthesesAgain } = \cfrac{9.49} { \left(\sqrt{29.11 }\right) } \nonumber \]

Then a square root:

\[ t_{SquareRoot } = \cfrac{9.49} {5.40} \nonumber \]

And then some division to finish it off!

\[ t_{TheEnd!} = 1.76 \nonumber \]

If your answer is close to 1.76 but not exactly (say, 1.73ish to 1.79ish), you probably did the equation correctly! The difference may be based on rounding; Dr. MO uses spreadsheets to do her calculations, rather than a calculator and hand-writing each difference score; spreadsheets keep hundreds of numbers after the decimal point, rather than just two, which affects the two numbers after the decimal point at the end of the calculation.

Step 4: Conclusion

The Conclusion step is about deciding whether or not to support the research hypothesis. To get to that, we first need to decide if the null hypothesis should be retained or rejected. And to get to that, we need to find the critical value so that we can compare the critical value to the t-score that you just calculated. Our critical values will once again be based on our level of significance, which we know is \(α\) = 0.05, the directionality of our test, which is one-tailed, and our degrees of freedom.

Exercise \(\PageIndex{3}\)

For our dependent-samples \(t\)-test, what is the critical t-score from the same t-table from when we learned about one-sample t-tests (or going to the Common Critical Values Table page)?

Answer

For our dependent-samples \(t\)-test, the degrees of freedom are still given as \(df = N – 1\) in which N is the number of pairs.

Degrees of Freedom \( = N-1 = 10 - 1 = 9 \)

With a Degrees of Freedom of 9, the critical t-score is 1.833 from the critical t-score table.

The critical t-score from the table (based on either decision rule) from Step 2 is larger than the calculated test statistic of \(t = 1.76\).

\((Critical < |Calculated|) =\) Reject null \(=\) means are different \(= p<.05\)

\((Critical > |Calculated|) =\) Retain null \(=\) means are similar \(= p>.05\)

Exercise \(\PageIndex{5}\)

1. Based on the critical value from the table and the calculated t-score, should we reject the null hypothesis?
2. What is the statistical sentence for this scenario's results?
3. Does this mean that the means are similar or different?

Answer

\(Critical (1.833) > |Calculated (1.76)| = \) Retain null \(=\) means are similar \(= p>.05\)

1. We retain (fail to reject) the null hypothesis.
2. t(9)=1.76,p>.05
3. The means are statistically similar to each other.

We retained the null hypothesis; now what?

Exercise \(\PageIndex{6}\)

Based on the means, our research hypothesis, and the statistical sentence is the research hypothesis supported or not supported? Why or why not?

Answer

The average Mindset Quiz score on the pretest was 40.70 points and the posttest was 43.70 points. Our research hypothesis was that the average score will be higher on the Mindset Quiz at the end of the semester (Time 2, posttest) than the average score at the beginning of the semester (Time 1, pretest). This is not supported because the statistical sentence of t(9)=1.76,p>.05 tells us that the means are similar (the probability that the null hypothesis is true in the population based on these samples is higher than our cut-off of 5% chance).

Step 5: Write-Up

What should the write-up look like? This write-up will sound sorta weird because the Posttest mean is higher than the Pretest mean, but the significance test shows that the means are similar.

Example \(\PageIndex{1}\)

Write up a conclusion for the mindset data.

Solution

I hypothesized that the average Mindset Quiz Posttest (\(\overline{X} = 43.70 \)) would be higher than the average Mindset Quiz Pretest (\(\overline{X} = 40.70 \)) after the journaling intervention. This research hypothesis is not supported; the means are not different enough (too similar) to state that they are from different populations (t(9)=1.76,p>.05). It appears that the journaling intervention did not statistically improve mindset scores for this sample of 10 community college students in an English course below college-level.

How are you feeling?

If you didn't quite get it yet, keep trying! Maybe check out the Learning (Statistics) section the very first chapter?

And when you do get the right calculation, interpret it correctly, and have a complete write-up, celebrate!