Question

1. A group of 24 participants (12 with knee injuries and 12 without) were randomly selected to participate in a study to investigate balance between those with knee 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 knee 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:

INJURY GROUP	83.1	82.6	90.9	75.3	83.6	89.5	77.1	74.7	80.3	82.9	88.0	84.3
NO INJURY GROUP	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 knee 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 knee injuries and for those without injuries.

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

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

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

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

2. Suppose a random group of 8 patients from Question 1 who had knee 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:

1 2 3 5 6 7 Patient injured leg score uninjured leg score 86.3 89.7 85.1 86.0 79.2 80.4 4. 83.7 80.1 82.6 82.6 87.3 87.0 88.1 89.7 8 89.5 92.4

For patients with knee 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) Define the parameter(s) of interest using the correct notation. Then, state the null and alternative hypotheses for this study.

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

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

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

Answer 1

Question 1:

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

The parameter(s) of interest is, µ, average score on the “Y Balance Test”.

The hypothesis being tested is:

H₀: µ₁ = µ₂

H₁: µ₁ < µ₂

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

The t-distribution is appropriate here.

t = -2.394

df = 22

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

p-value = 0.0128

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

Since the p-value (0.0128) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that patients with knee injuries have a lower average score on the “Y Balance Test” than those without injuries.

INJURY GROUP	NO INJURY GROUP
82.692	88.208	mean
5.231	6.029	std. dev.
12	12	n
	22	df
	-5.5167	difference (INJURY GROUP - NO INJURY GROUP)
	31.8563	pooled variance
	5.6441	pooled std. dev.
	2.3042	standard error of difference
	0	hypothesized difference
	-2.394	t
	.0128	p-value (one-tailed, lower)