Evan Kurnia's copy of Perform an independent two sample t-test

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Evan Kurnia Alim 6 years, 4 months ago

Created this as a copy of Perform an independent two sample t-test.

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 by hand (or by entering the data and using SPSS) 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	3.590
sd2	number	4.422
m1	number	16
tol	number	0.001
m2	number	12
msg	list	List of 5 items
cmsg1	list	List of 5 items
t999	number	3.922
atvalue	number	2.2205243073
sig1	number	4.82
tvalue	number	2.2205243073
sig2	number	3.07
t99	number	2.878
t95	number	2.101
t90	number	1.734
cmsg	list	List of 5 items
mu1	number	15.7
r1	list	List of 10 items
r2	list	List of 10 items
mu2	number	11.25
pmsg	list	List of 5 items
s	number	4.028
t	integer	2
v	list	[ 0, 0, 1, 0, 0 ]
n1	integer	10
n2	integer	10
pvalue	number	0.0394584874

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Name

Data type

Value

xxxxxxxxxx
 
precround(pstdev(r1),3)

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Description

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

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Generated value: `number`

3.590

← Depends on:

→ Used by:

Advice
"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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1. Step Information only
Part b) Gap-fill
1. Gap Number entry
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Part c) Choose one from a list
Part d) Choose one from a list
Part e) Choose one from a list

Gap-fill

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:

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

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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Evan Kurnia's copy of Perform an independent two sample t-test

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Evan Kurnia Alim 6 years, 4 months ago

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Evan Kurnia's copy of Perform an independent two sample t-test

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Evan Kurnia Alim 6 years, 4 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`