Perform an independent two sample t-test

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Pete Philipson	said	Needs to be tested	1 year, 5 months ago
Chris Graham	said	Has some problems	2 years, 3 months ago

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Pete Philipson 1 year, 5 months ago

Gave some feedback: Needs to be tested

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Chris Graham commented 2 years, 3 months ago

Part b) asks for the test statistic, which is used as part of the adaptive marking in part a), so part (a) ends up pending with the "This feedback is based on your last submitted answer. Submit your changed answer to get updated feedback" message after submitting part b). It caused some confusion when students moved between questions in the exam, as they were presented with the warning that not all questions were submitted.

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Chris Graham 2 years, 3 months ago

Gave some feedback: Has some problems

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Tom Lowe commented 4 years, 4 months ago

I've corrected some typos, moved the steps that tell you how to calculate the test statistic to the part that asks you to calculate it, made the steps show more clearly that the absolute value must be taken, and changed the test statistic formula in the advice so that it matches the one in the steps. Question now seems fine

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Tom Lowe 4 years, 4 months ago

Saved a checkpoint:

Have corrected some typos, and am about to move the step showing how to calculate the test statistic from part a to part b (where the test statistic question is asked) but want to save first in case I accidentally break it.

Restore this checkpoint

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Chris Graham 6 years, 5 months ago

Gave some feedback: Needs to be tested

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Chris Graham 6 years, 5 months ago

Created this.

Name	Status	Author	Last Modified
Perform an independent two sample t-test	Needs to be tested	Chris Graham	18/01/2024 14:30
Evan Kurnia's copy of Perform an independent two sample t-test	draft	Evan Kurnia Alim	21/11/2018 12:45

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Question statement

The quadriceps angle, or Q angle, is the angle formed by the encounter of two lines, one that starts at the anterior iliac spine (AIS) and goes to the center of the patella, and another that goes from the tibial tuberosity to the center of the patella. The following table shows the Q-angles, in degrees, for a random sample of 10 males and 10 females.

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

Carry out a two-sample $t$ -test to test if there is evidence of a difference in the average Q-angle between males and females.

Let $\mu_1$ be the mean Q angle for females.

Let $\mu_2$ be the mean Q angle for males.

Null hypothesis: $H_0$ : $\mu_1 = \mu_2$

Alternative hypothesis: $H_1$ : $\mu_1 \ne \mu_2$

Give any introductory information the student needs.

			Name	Type	Generated Value

sd1	number	4.894
sd2	number	2.998
m1	number	15.2
tol	number	0.001
m2	number	13.1
msg	list	List of 5 items
cmsg1	list	List of 5 items
t999	number	3.922
atvalue	number	1.1571569129
sig1	number	4.74
tvalue	number	1.1571569129
sig2	number	3.5
t99	number	2.878
t95	number	2.101
t90	number	1.734
cmsg	list	List of 5 items
mu1	number	15.48
r1	list	List of 10 items
r2	list	List of 10 items
mu2	number	11.98
pmsg	list	List of 5 items
s	number	4.058
t	integer	4
v	list	[ 0, 0, 0, 0, 1 ]
n1	integer	10
n2	integer	10
pvalue	number	0.2623341669

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precround(pstdev(r1),3)

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4.894

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

Find the mean and standard deviations of the Q-angles for the males and females:

Sample mean Q-angle for males = Gap 2.

Sample standard deviation of Q-angles for males = Gap 3.

Sample mean Q-angle for females = Gap 0.

Sample standard deviation of Q-angles for females = Gap 1.

Advice

We test the null hypothesis

$H_0:\; \mu_1=\mu_2$ (there is no significant difference in the Q-angles of males and females)

against the alternative hypothesis

$H_1:\; \mu_1 \neq \mu_2$ (there is a significant difference in the Q-angles of males and females).

We find that the mean Q-angle of the females is $\overline{x}_1=\var{m1}$ with standard deviation $s_1=\var{sd1}$ and the mean Q-angle of the males is $\overline{x}_2=\var{m2}$ with standard deviation $s_2=\var{sd2}$.

We also have that $n_1=n_2=10$ and so we can calculate the pooled standard deviation $s$:

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

We can now calculate the test statistic:

\[\begin{eqnarray*}T&=& \frac{\lvert \;\overline{x}_1-\overline{x}_2\;\rvert}{s\;\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\&=&\frac{\lvert\var{m1}-\var{m2}\rvert}{\var{s}\sqrt{\frac{1}{\var{n1}}+\frac{1}{\var{n2}}}}\\&=&\var{tvalue}\end{eqnarray*}\].

Remeber that we want the absolute value of this result - that is, we always take it to be positive.

We then look up $n_1+n_2-2=18$ degrees of freedom in the tables and see where our test statistic lies.

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

We see that the test 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 Q-angle in females and males.

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Pete Philipson 1 year, 5 months ago

Chris Graham commented 2 years, 3 months ago

Chris Graham 2 years, 3 months ago

Tom Lowe commented 4 years, 4 months ago

Tom Lowe 4 years, 4 months ago

Chris Graham 6 years, 5 months ago

Chris Graham 6 years, 5 months ago

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Perform an independent two sample t-test

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Pete Philipson 1 year, 5 months ago

Chris Graham commented 2 years, 3 months ago

Chris Graham 2 years, 3 months ago

Tom Lowe commented 4 years, 4 months ago

Tom Lowe 4 years, 4 months ago

Chris Graham 6 years, 5 months ago

Chris Graham 6 years, 5 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`