Question

1. The anterior cruciate ligament (ACL) is one of the major ligaments in the knee. The ACL is especially important for many sports as it helps provide stability in the knee joint. A group of 24 participants (12 with ACL injuries and 12 without) were randomly selected to participate in a study to investigate balance between those with ACL injuries and those without. Each participant was asked to perform a “Y Balance Test” in which they stand on a leg in the squat position (in the case of the ACL injury group the participants stood on their injured leg) while reaching their other leg out as far as possible in each of three directions that are marked on the ground with tape in a “Y” shape. To control for variation in body sizes each participant had the measurements in each of the three directions compiled into a composite score that was then divided by the participants leg length to convert the measurement into a score out of 100. The results for the participants are recorded below:

ACLinjurygroup 83.1 82.6 90.9 75.3 83.6 89.5 77.1 74.7 80.3 82.9 88.0 84.3

noinjurygroup 86.4 92.3 91.7 95.2 98.0 87.2 89.4 76.5 79.4 87.8 85.9 88.7

Do patients with ACL injuries have a lower average score on the “Y Balance Test” than those without injuries? If needed, you may assume that performance on the “Y Balance Test” is normally distributed, both for patients with ACL injuries and for those without injuries.

(a) [2 marks] Define the parameter(s) of interest using the correct notation. Then, state the null and alternative hypotheses for this study.

(b) [1 mark] Calculate the observed value of the test statistic. State the distribution (and degrees of freedom if needed) it follows.

(c) [1 mark] Compute the p-value or provide a range of appropriate values for the p-value.

(d) [1 mark] Using the significance level α = 0.025, state your conclusions about performance on the “Y Balance Test” between the ACL injury group versus the group with no injuries.

2. Suppose a random group of 8 patients from Question 1 who had ACL injuries were asked to perform the “Y Balance Test”, however this time the patients performed the test once while standing on their injured leg, and then after a 5 minute rest period were asked to perform the test again while standing on their uninjured leg. The results are recorded below:

injured leg score 86.3 85.1 79.2 83.7 82.6 87.3 88.1 89.5

uninjured leg score 89.7 86.0 80.4 80.1 82.6 87.0 89.7 92.4

For patients with ACL injuries are average scores on the “Y Balance Test” different while standing on the injured leg versus the uninjured leg? If needed, you may assume that performance on the “Y Balance Test” is normally distributed on both injured and uninjured legs.

(a) [2 marks] Define the parameter(s) of interest using the correct notation. Then, state the null and alternative hypotheses for this study.

(b) [1 mark] Calculate the observed value of the test statistic. State the distribution (and degrees of freedom if needed) it follows.

(c) [1 mark] Compute the p-value or provide a range of appropriate values for the p-value.

(d) [1 mark] Using the significance level α = 0.10, state your conclusions about performance on the “Y Balance Test” for patients with ACL injuries on their injured versus uninjured legs.

(b) [1 mark] Calculate the observed value of the test statistic. State the distribution (and degrees of freedom if needed) it follows.

(c) [1 mark] Compute the p-value or provide a range of appropriate values for the p-value.

(d) [1 mark] Using the significance level α = 0.10, state your conclusions about performance on the “Y Balance Test” for patients with ACL injuries on their injured versus uninjured legs.

please answer question 2

Answer 1

The first question is to compare means for two independent samples. Independent because different subjects are used in the two data sets. Here we compare the difference of the overall population as a whole

The second question is to compare means for dependent samples. Dependent because same subjects are used in the two data sets. Here we compare individual difference and the test the population of this difference.

Q1

	With (1)	Without (2)	X12	X22
1	83.1	86.4	6905.61	7464.96
2	82.6	92.3	6822.76	8519.29
3	90.9	91.7	8262.81	8408.89
4	75.3	95.2	5670.09	9063.04
5	83.6	98	6988.96	9604
6	89.5	87.2	8010.25	7603.84
7	77.1	89.4	5944.41	7992.36
8	74.7	76.5	5580.09	5852.25
9	80.3	79.4	6448.09	6304.36
10	82.9	87.8	6872.41	7708.84
11	88	85.9	7744	7378.81
12	84.3	88.7	7106.49	7867.69
Total	992.3	1058.5	82355.97	93768.33

Mean =

Σ. .. n

SD =

Σ (Στ)2 ✓Var η η –1

	With (1)	Without (2)
Mean	82.6917	88.2083
Variance	27.3663	36.3463
n	12	12

(a) [2 marks] Define the parameter(s) of interest using the correct notation. Then, state the null and alternative hypotheses for this study.

Do patients with ACL injuries have a lower average score on the “Y Balance Test” than those without injuries? That is if (1) < (2): (1) - (2) < 0. We are comparing two population means.

parameter(s) of interest: The population means of the scores of 'Y Balance Test'

  
11
and
112

test

$H_{0}:\mu1=\mu2$:

$H_{1}:\mu1<\mu2$

one left sided test

(b) [1 mark] Calculate the observed value of the test statistic. State the distribution (and degrees of freedom if needed) it follows.

assuming the population variances are equal.

State the distribution (and degrees of freedom if needed): The data is assumed to be normal and the population variances are not given so we will use independent samples t-test for difference of population means.

df = 22 (n1+ n2 -2)

Pooled variance =

Vari(nı – 1) + Varz(n2 - 1) ni + n2 - 2

= 31.8563

Test Stat =

(21-12) – (ui - U2) Pooled Var(1/n1 + 1/n2)

Test Stat = -2.3942

(c) [1 mark] Compute the p-value or provide a range of appropriate values for the p-value.

p-value = P ( > |T.S.| )

tn +n2-2

= P(t₂₂ > 2.39)

p-value = 0.0128 ................using t-dist tables

(d) [1 mark] Using the significance level α = 0.025, state your conclusions about performance on the “Y Balance Test” between the ACL injury group versus the group with no injuries.

Since p-value < 0.025

we reject the null hypothesis at 2.5%. There is sufficient evidence to conclude that the patients with ACL injuries have a lower average score on the “Y Balance Test” than those without injuries.

experts are complusorily to solve only 1 question.