Independent two sample t-test.

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Bill Foster

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Bill Foster 12 years, 2 months ago

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

Carry out by hand a two-sample t-test to test 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

sd1	number	7.156
sd2	number	8.556
m1	number	58.1
tol	number	0.001
m2	number	70.1
msg	list	List of 5 items
cmsg1	list	List of 5 items
t999	number	3.922
atvalue	number	3.402
sig1	number	10
tvalue	number	-3.402
sig2	number	9
t99	number	2.878
t95	number	2.101
t90	number	1.734
cmsg	list	List of 5 items
mu1	number	58.5
r1	list	List of 10 items
r2	list	List of 10 items
mu2	number	70
pmsg	list	List of 5 items
s	number	7.887
t	integer	1
v	list	[ 0, 1, 0, 0, 0 ]
n1	integer	10
n2	integer	10
pvalue	number	0.0031774037

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precround(pstdev(r1),3)

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Generated value: `number`

7.156

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"Part a)" → "Gap 1." - Minimum accepted value
"Part a)" → "Gap 1." - Maximum accepted value

Parts

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

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Find the mean and standard deviations of the scores of the two groups:

Mean scores of Group 1= Gap 0. (input to 3 decimal places)

Standard deviation of scores for Group 1 = Gap 1. (input to 3 decimal places)

Mean scores of Group 2= Gap 2. (input to 3 decimal places)

Standard deviation of scores for Group 2 = Gap 3. (input to 3 decimal places)

Now find the two sample t-test statistic $T$ using the values you have just calculated to 3 decimal places and input $|T|$ here: Gap 4. (3 decimal places)

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 $\overline{x}_1=\var{m1}$ with standard deviation $s_1=\var{sd1}$ and the mean score of Group 2 is $\overline{x}_2=\var{m2}$ 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:

\[\begin{eqnarray*}T&=& \frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{s\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\&=&\frac{(\var{m1}-\var{m2})-(0)}{\var{s}\sqrt{\frac{1}{\var{n1}}+\frac{1}{\var{n2}}}}\\&=&\var{tvalue}\end{eqnarray*}\] to 3 decimal places.

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

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

\[\begin{array}{r|rrrrr}&0.20&0.10&0.05&0.01&0.001\\\hline18&1.330&1.734&2.101&2.878&3.922\end{array}\]

We see that the t-statistic {msg[t]} and the table tells us that the $p$ value {pmsg[t]}.

Hence we conclude that we {cmsg[t]} the null hypothesis. There is {cmsg1[t]} evidence of a difference between the average scores of the two groups.

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Bill Foster 12 years, 2 months ago

Error in variable testing condition

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Independent two sample t-test.

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 12 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`