Marlon's copy of Adaptive marking: Independent two sample t-test

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Marlon's copy of Adaptive marking: Independent two sample t-test

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Marlon's copy of Adaptive marking: Independent two sample t-test

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Taxonomy: mathcentre

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Taxonomy: Context

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Generated value: `integer`

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Generated value: integer

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Generated value: `integer`

Marlon Arcila 8 years, 9 months ago

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Marlon Arcila 8 years, 9 months ago

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Marlon Arcila 8 years, 9 months ago

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Marlon Arcila

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Created this as a copy of Adaptive marking: Independent two sample t-test.

Name	Status	Author	Last Modified
Adaptive marking: Independent two sample t-test	Ready to use	Christian Lawson-Perfect	19/12/2022 12:16
Marlon's copy of Adaptive marking: Independent two sample t-test	draft	Marlon Arcila	18/07/2016 17:47
STAT7008: Independent two sample t-test	draft	Catherine Palmer	25/11/2018 22:07
Two sample t-test: question order and scores	Ready to use	Mario Orsi	29/01/2025 10:24
Ulrich's copy of Adaptive marking: Independent two sample t-test	draft	Ulrich Görtz	15/02/2021 20:06

There are 12 other versions that do you not have access to.

Question statement

An educational psychologist claimed that the order in which questions were asked affected the student’s ability to answer them correctly and hence their total score. In order to test this, $20$ students were randomly divided into two groups of $10$ . The first group were given questions in increasing order of difficulty and the second group in decreasing order of difficulty. The ordered test scores obtained were:

Group 1	{r1[0]}	{r1[1]}	{r1[2]}	{r1[3]}	{r1[4]}	{r1[5]}	{r1[6]}	{r1[7]}	{r1[8]}	{r1[9]}
Group 2	{r2[0]}	{r2[1]}	{r2[2]}	{r2[3]}	{r2[4]}	{r2[5]}	{r2[6]}	{r2[7]}	{r2[8]}	{r2[9]}

Carry out a two-sample t-test to decide if there is evidence of a difference in the average test scores for the two sets of students.

Give any introductory information the student needs.

			Name	Type	Generated Value

n1	integer	10
n2	integer	10
mu1	number	60
sigma1	number	8.8
mu2	number	71.5
sigma2	number	10

			Name	Type	Generated Value

			r1	list	List of 10 items
			r2	list	List of 10 items

			Name	Type	Generated Value

mean1	number	60
sd1	number	8.459
mean2	number	69.6
sd2	number	10.967
s	number	9.794
t_statistic	number	-2.1917758407
p_value	number	0.0417873982

			Name	Type	Generated Value

			Name	Type	Generated Value

t90	number	1.734
t95	number	2.101
t99	number	2.878
t999	number	3.922

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
10

JME function reference

Description

Size of sample 1

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

This variable doesn't seem to be used anywhere.

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
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5. Gap Gap 4. Number entry
  Add
1. Step Information only
Part b) Choose one from a list
Part c) Choose one from a list
Part d) Choose one from a list

Ask the student a question, and give any hints about how they should answer this part.

Find the mean and standard deviations of the scores of the two groups. Round your answers to 3 decimal places.

	Mean	Standard deviation
Group 1	Gap 0.	Gap 1.
Group 2	Gap 2.	Gap 3.

Now find the two sample t-test statistic $T$ using the values you have just calculated and enter it here: Gap 4.

Advice

We test the following hypothesis,

$H_0:\; \mu_1=\mu_2$ versus $H_1:\; \mu_1 \neq \mu_2$

We find that the mean score of Group 1 is $\overline{x}_1=\var{mean1}$ with standard deviation $s_1=\var{sd1}$ and the mean score of Group 2 is $\overline{x}_2=\var{mean2}$ with standard deviation $s_2=\var{sd2}$.

(All calculated to 3 decimal places.)

Using the formula for the two-sample $t$-statistic as shown above with $n_1=n_2=10$:

The estimate of the pooled variance is calculated to be:

\[s^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}= \frac{\var{n1-1}\times \var{sd1}^2+\var{n2-1}\times \var{sd2}^2}{\var{n1+n2-2}}=\var{s^2}.\]

Hence $s = \sqrt{\var{s^2}}=\var{s}$ to 3 decimal places.

We find that the t-statistic has value:

\begin{align}
T &= \frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{s\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \\
&= \frac{(\var{mean1}-\var{mean2})-(0)}{\var{s}\sqrt{\frac{1}{\var{n1}}+\frac{1}{\var{n2}}}} \\
&= \var{t_statistic}
\end{align}

Our test statistic is $|T|=\var{abs(t_statistic)}$.

Given that we have $n_1+n_2-2=18$ degrees of freedom, we look up this value on the T-distribution table for $t_{18}$

\[\begin{array}{r|rrrrr}&0.10&0.05&0.01&0.001\\\hline18&1.734&2.101&2.878&3.922\end{array}\]

We see that the t-statistic {t_statistic_range} and the table tells us that the $p$ value {p_value_range}.

Hence we conclude that we {reject} the null hypothesis. There is {evidence_strength} evidence of a difference between the average scores of the two groups.

Give a worked solution to the whole question.

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There was an error which means the tests can't run:

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This question is not used in any exams.

Error evaluating variable scenario: No definition of 'sum' of correct type found.

decision_marking_matrix

Error evaluating variable decision_marking_matrix: Variable scenario is not defined.

Error evaluating variable reject: Variable scenario is not defined.

evidence_strength

Error evaluating variable evidence_strength: Variable scenario is not defined.

t_statistic_range

Error evaluating variable t_statistic_range: Variable scenario is not defined.

p_value_range

Error evaluating variable p_value_range: Variable scenario is not defined.