Question

Prednisolone is a relatively new drug for treating inflammatory diseases. This drug replaces prednisone, a corticosteroid that can cause patients to develope osteoporosis, low bone density if the drug is used for a prolonged time period. Here, we are comparing a group of 30 cats receiving prednisolone to a group of 30 cats receiving a placebo. The cats receiving a placebo will serve as a control group. After receiving treatment for 6 weeks a bone scan was done to assess bone density in cats receiving prednisolone and cats receiving a placebo.

H₀: Cats receiving prednisolone will not have a different bone density than cats receiving a placebo.

H_A: Cats receiving prednisolone will have a different bone density than cats receiving a placebo.

Rep #	Control	Prednisolone
1	0.5	0.52
2	0.49	0.4
3	0.51	0.52
4	0.47	0.44
5	0.49	0.39
6	0.51	0.4
7	0.53	0.53
8	0.53	0.59
9	0.53	0.58
10	0.5	0.47
11	0.56	0.52
12	0.5	0.49
13	0.47	0.41
14	0.45	0.47
15	0.47	0.48
16	0.49	0.44
17	0.51	0.5
18	0.53	0.53
19	0.46	0.55
20	0.54	0.38
21	0.49	0.57
22	0.41	0.46
23	0.46	0.48
24	0.55	0.43
25	0.57	0.48
26	0.45	0.31
27	0.49	0.48
28	0.55	0.52
29	0.48	0.53
30	0.59	0.5

If decimals are part of the answer, round your answers to two significant figures unless otherwise specified.

When calculating a t-value, you should arrive at a positive value. This is accomplished by setting N₁ = to the larger mean and N₂ = to the smaller mean.

Here is the equation for a two tailed unpaired t-test. Use excel or sheets to calculate the components of the equation and then use a formula in excel or sheets to do all the arithmetic in one step. Do not round until the very end.

What is the average bone density of cats receiving a placebo? _________ g/cm³

What is the average bone density of cats receiving prednisolone? ___________ g/cm³

What is N for the control group? __________

What is N for the prednisolone group? ___________

What is the standard deviation for the control group? __________

What isstandard deviationfor the prednisolone group? ____________

How many degrees of freedom does this experiment have? __________

What will the numerator for the t-test equal? ___________

What will the denominator for the t-test equal? __________

What is the calculated t value for comparing cats treated with prednisolone to cats receiving a placebo? __________? (round to 2 decimals)

Use the t-table below to answer the following question. Use an alpha value of 0.05 for a two-tailed t-test.

Will you reject or fail to reject the null hypothesis that prednisalone does not have an effect on bone density? __________?

Answer 1

Sol:

So sample sizes:

$n_{1}=n_{2}=30$

The mean is

$\bar{x}_{1}=\frac{\sum x_{1}}{n_{1}}=\frac{15.08}{30}=0.5027$

$\bar{x}_{2}=\frac{\sum x_{2}}{n_{2}}=\frac{14.37}{30}=0.4790$

The standard deviation is

$s_{1}=\sqrt{\frac{\sum \left ( x_{1}-\bar{x}_{1} \right )^{2}}{n_{1}-1}}=0.0402$

$s_{2}=\sqrt{\frac{\sum \left ( x_{2}-\bar{x}_{2} \right )^{2}}{n_{2}-1}}=0.0646$

Since it is not given that variances are equal so degree of freedom of the test is

$df=\frac{\left ( \frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}} \right )^{2}}{\frac{\left ( s_{1}^{2}/n_{1} \right )^{2}}{n_{1}-1}+\frac{\left ( s_{2}^{2}/n_{2} \right )^{2}}{n_{2}-1}}=48$

So df is 48.

And test statistics will be

$t=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}=1.70$

From t tbale p-value lies between 0.05<p<0.10.

Since p-value is greater than 0.05 so we fail ot reject the null hypothesis.