Question

Homework assistance for accounting students. How much assistance should accounting professors provide students for completing homework? Is too much assistance counterproductive? These were some of the questions of interest in a Journal of Accounting Education (Vol. 25, 2007) article. A total of 75 junior-level accounting majors who were enrolled in Intermediate Financial Accounting participated in an experiment. All students took a pretest on a topic not covered in class; then, each was given a homework problem to solve on the same topic. However, the students were randomly assigned different levels of assistance on the homework. Some (20 students) were given the completed solution, some (25 students) were given check figures at various steps of the solution, and the rest (30 students) were given no help. After finishing the homework, each student was given a posttest on the subject. One of the variables of interest to the researchers was the knowledge gain (or test score improvement), measured as the difference between the posttest and pretest scores. The sample means knowledge gains for the three groups of students are provided in the table.

a. The researchers theorized that as the level of homework assistance increased, the test score improvement from pretest to post test would decrease. Do the sample means reported in the table support this theory?

b. What is the problem with using only the sample means to make inferences about the population mean knowledge gains for the three groups of students?

c. The researchers conducted a statistical test of the Hypothesis to compare the mean knowledge gain of students in the "no solutions" group with the mean knowledge gain of students in the "check figures" group. Based on the theory, part a sets up the null and alternative hypotheses for the test.

d. The observed significance level of the t-test of the part$\mathit{c}$ was reported as$\mathbf{8248}$ Using $\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$, interpret this result.

e. The researchers conducted a statistical test of the hypothesis to compare the mean knowledge gain of students in the "completed solutions" group with the mean knowledge gain of students in the "check figures" group. Based on the theory, part a sets up the null and alternative hypotheses for the test.

f. The observed significance level of the role="math" localid="1652694732458" $\mathit{t}$-test of part e was reported as 1849. Using $\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$, interpret this result.

g. The researchers conducted a statistical test of the Hypothesis to compare the mean knowledge gain of students in the "no solutions" group with the mean knowledge gain of students in the "completed solutions" group. Based on the theory, part a sets up the null and alternative hypotheses for the test.

h. The observed significance level of the $\mathbf{t}$-test of the part was$\mathbf{g}$ reported as$2726.$ Using role="math" localid="1652694677616" $\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$, interpret this result.

Short answer

1. No, the researchers' theory is not correct.
2. The sample mean is insufficient for making inferences about the population mean knowledge gains for the three groups.
3. The hypotheses are given below:

Null Hypothesis:

${\mathrm{H}}_0:({\mathrm{\mu }}_1-{\mathrm{\mu }}_2)=0$

Alternative Hypothesis:

${\mathrm{H}}_1:({\mathrm{\mu }}_1-{\mathrm{\mu }}_2)>0$

d. There is a preliminary result that the mean knowledge gained by students in the "no solution" group is greater than the mean knowledge gained in the "check figure" group.

e. The hypotheses are given below:

Null Hypothesis:

${\mathrm{H}}_0:({\mathrm{\mu }}_2-{\mathrm{\mu }}_3)=0$

Alternative Hypothesis:

${\mathrm{H}}_1:({\mathrm{\mu }}_2-{\mathrm{\mu }}_3)>0$

f. There is a preliminary result that the mean knowledge gained by students in the "check figure" group is greater than the mean knowledge gained in the "complete solution" group.

g. The hypotheses are given below:

Null Hypothesis:

${\mathrm{H}}_0:({\mathrm{\mu }}_1-{\mathrm{\mu }}_3)=0$

Alternative Hypothesis:

${\mathrm{H}}_1:({\mathrm{\mu }}_1-{\mathrm{\mu }}_3)>0$

Step-by-step solution

Step-by-Step SolutionStep 1: Given information

One survey was done on Homework assistance for accounting students. Seventy-five junior-level accounting majors were taken as samples. Among them, 20 students were given complete solutions, 25 students were given check figures at various solution steps, and the remaining 30 students were given no solution. Two tests were conducted before giving these things and after giving this solution. The means are given for three groups.

(a) Do the sample mean reported in the table support this theory

Here, the sample mean of the "no solution" group is $2.43$, the sample mean of the "Check figures" group is $2.72$, and the sample mean of the "Complete solution" group is $1.95$.

From "no solution" to "Check solution," the sample mean increased, but from "Check solution" to "complete solution," the sample mean decreased.

So, from checking the mean, it cannot be said that the researchers' theory "as a level of homework assistance increased, the test score improvement from pretest to posttest would decrease" is not correct.

(b) For making the inference, why only mean is not sufficient.

Only from the mean is it not possible to know about the variability or the spread of the probability distribution of the population.

So, the sample mean is insufficient for making inferences about the population mean knowledge gains for the three groups.

(c) Null and Alternative Hypothesis 

Let$\mathrm{\mu }$is the mean knowledge gained for the student in the "no solution" group and the mean knowledge gained for the student in the "check figure" group.

The Hypothesis is given by:

Null Hypothesis:

${\mathrm{H}}_0:({\mathrm{\mu }}_1-{\mathrm{\mu }}_2)=0$.There is no significant difference between the mean knowledge gained for the two groups of students.

Alternative Hypothesis:

${\mathrm{H}}_1:({\mathrm{\mu }}_1-{\mathrm{\mu }}_2)>0$There is evidence to say the mean knowledge gained by students in the "no solution" group is greater than the mean knowledge gained in the "check figure" group.

(d) Interpreting the result based on the observed significance level of t- the test (pvalue)

The significance level of $t$the test is $0.8248$and $\alpha =0.05$.

If role="math" localid="1652695802085" $p$-value role="math" localid="1652695808508" $<\alpha$, then reject the null Hypothesis role="math" localid="1652695811737" ${\mathrm{H}}_0$.

If role="math" localid="1652695798375" $p$-value, role="math" localid="1652695805481" $>\alpha$then fail to reject the null Hypothesis role="math" localid="1652695814880" ${\mathrm{H}}_0$.

From the given information, the role="math" localid="1652695886251" $p$-value $=0.8248$

Conclusion:

Here,$p$-the value is greater than the level of significance.

That is,$p$-value $(=0.8248)>\alpha (=0.05)$.

The reject criterion does not deny the Null Hypothesis.

Thus, it can be concluded that there is insufficient result to say that the mean knowledge gain of students in the "no solution" group is greater than the mean knowledge gain of students in the "check figure" group $\alpha =0.05$.

(e) Null and Alternative Hypothesis

Let${\mathrm{\mu }}_2$is the mean knowledge gained for the student in the "check figure" group and the mean knowledge gained for the student in the "complete solution" group.

The Hypothesis is given by:

Null Hypothesis:

${\mathrm{H}}_0:({\mathrm{\mu }}_2-{\mathrm{\mu }}_3)=0$

There is no significant difference between mean knowledge gains for the two groups of students.

Alternative Hypothesis:

${\mathrm{H}}_1:({\mathrm{\mu }}_2-{\mathrm{\mu }}_3)>0$

There is evidence to say the mean knowledge gained by students in the "check figure" group is greater than the mean knowledge gained in the "complete solution" group.

(f) Interpreting the result based on the observed significance level of t-test ( pvalue)

The significance level of$t$the test is$0.1849$and$\alpha =0.05$.

Calculation:

Decision:

If $p$-value $<\alpha$, then reject the null Hypothesis$H_0$.

If role="math" localid="1652696409921" $p$-value role="math" localid="1652696413312" $>\alpha$, then fail to reject the null Hypothesis $H_0$.

From the given information, the role="math" localid="1652696346800" $p$-value role="math" localid="1652696351540" $=0.1849$

Conclusion:

Here,$p$-the value is greater than the level of significance.

That is, $p$- value $(=0.1849)>\alpha (=0.05)$.

The reject criterion does not deny the Null Hypothesis.

Thus, it can be concluded that there is insufficient result to say that the mean knowledge gain of students in the "check figure" group is greater than the mean knowledge gain of students in the "complete solution" group.

(g) Null and Alternative Hypothesis

The Hypothesis is given by:

Null Hypothesis:

${\mathrm{H}}_0:({\mathrm{\mu }}_1-{\mathrm{\mu }}_3)=0$

There is no significant difference between mean knowledge gains for the two groups of students.

Alternative Hypothesis:

role="math" localid="1652696508287" ${\mathrm{H}}_1:({\mathrm{\mu }}_1-{\mathrm{\mu }}_3)>0$

There is evidence to say the mean knowledge gained by students in the "no solution" group is greater than the mean knowledge gained in the "complete solution" group.

(h) Interpreting the result based on the observed significance level of t-test (  pvalue)

There is a preliminary result to say that the mean knowledge gained by students in the "no solution" group is greater than the mean knowledge gained in the "complete solution" group.

The significance level of $t$the test is $0.2726$and role="math" localid="1652696894149" $\alpha =0.05$.

Ifrole="math" localid="1652696873877" $p$-valuerole="math" localid="1652696902174" $<\alpha$, then reject the null Hypothesis$H_0$.

Ifrole="math" localid="1652696870754" $p$-value,$>\alpha$then fail to reject the null Hypothesis$H_0$.

From the given information, the role="math" localid="1652696801227" $p$-value role="math" localid="1652696806196" $=0.2726$

Conclusion:

Here,role="math" localid="1652696865840" $p$-the value is greater than the level of significance.

That is, role="math" localid="1652696862302" $p$- value $(=0.2726)>\alpha (=0.05)$.

The reject criterion does not deny the Null Hypothesis.

Thus, it can be concluded that there is insufficient result to say that the mean knowledge gain of students in the "no solution" group is greater than the mean knowledge gain of students in the "complete solution" group.