Question

1. Researchers examined the effect of phosphate supplementation on bone formation in 6 healthy adult dogs. For each dog, bone formation was measured twice: once after 12 weeks of phosphate supplementation, and once after a 12-week control period. The results in percent bone growth per year are shown below. Do these data provide evidence that phosphate supplementation significantly stimulates bone formation?

Dog ID	1	2	3	4	5	6
Control	1.73	3.37	3.59	2.05	1.86	3.60
Phosphate	8.16	4.58	3.98	5.24	3.04	7.03

What are the appropriate null and alternative hypotheses?

1. H₀:m_Control = m_Phosphate versus H_a: m_Control < m_Phosphate
2. H₀: m_{Difference control–phosphate} = 0 versus H_a:m_{Difference control–phosphate} < 0
3. H₀:m_Control = m_Phosphate versus H_a: m_Control ≠ m_Phosphate
4. H₀:m_{Difference control–phosphate} = 0 versus H_a: m_{Difference control–phosphate} ≠ 0

2. What is the test statistic for the appropriate test (show calculation)?

1. z= –2.64
2. z= –2.92
3. t= –2.92
4. t= –3.04

3. What is the P-value for the appropriate test? Specify the distribution used and all relevant parameters.

4. True or False: A large P-value implies that H₀is necessarily true,
   1. True
   2. False

Answer 1

Appropriate null and alternative hypothesis is

a) Ho:mcontrol=mphospate

H1:mcontrol < mphospate

Let

X:Bone formation in control period

Y:Bone formation when phospate supplement is given

Now as this is 2 sample independent t-test

The test statistic is

Where s1^2 is the sample variance of X

s2^2 is sample variance of Y

n1=n2=6

So mean of X and mean of Y are given by

Similarly

Therefore the t statistic is

Hence option d.

The p-value is given by

where T follows t-distribution

p-value therefore is equal to 0.009287

As the p-value is less than 0.05 at 5% LOS we reject Null hypthesis which implies that the mcontrol <mphospate.

True, large p-value implies Ho is necessarily true.