Question

The Merck Manual states that, for healthy adults, the mean number of milliliters of oxygen per deciliter of blood is 19.0. A company that sells vitamins claims that its multivitamin complex will increase the oxygen capacity of the blood. A random sample of 28 adults took the vitamin for six months. After blood tests, it was found that the sample mean was 20.7 ml of oxygen per deciliter of blood with a standard deviation of 6.7ml.

a. At the 0.05 level, test the claim that the average oxygen capacity has increased.

b. How much power do you have to detect a 3ml difference (from the null) in the average amount of oxygen in the blood?

c. What sample size would you need to have 90% power to detect this observed difference?

Answer 1

The Merck Manual states that, for healthy adults, the mean number of milliliters of oxygen per deciliter of blood is 19.0. A company that sells vitamins claims that its multivitamin complex will increase the oxygen capacity of the blood. A random sample of 28 adults took the vitamin for six months. After blood tests, it was found that the sample mean was 20.7 ml of oxygen per deciliter of blood with a standard deviation of 6.7ml.

MINITAB used

a. At the 0.05 level, test the claim that the average oxygen capacity has increased.

One-Sample T

Descriptive Statistics

N	Mean	StDev	SE Mean	95% Lower Bound for μ
28	20.70	6.70	1.27	18.54

μ: mean of Sample

Test

Null hypothesis	H₀: μ = 19
Alternative hypothesis	H₁: μ > 19

T-Value	P-Value
1.34	0.095

Calculated t=1.34, P=0.095 which is > 0.05 level of significance. Ho is not rejected.

There is not enough evidence to support the claim that the average oxygen capacity has increased.

b. How much power do you have to detect a 3ml difference (from the null) in the average amount of oxygen in the blood?

Power and Sample Size

1-Sample t Test

Testing mean = null (versus > null)

Calculating power for mean = null + difference

α = 0.05 Assumed standard deviation = 6.7

Results

Difference	Sample Size	Power
3	28	0.746794

The attained power = 0.7468

c. What sample size would you need to have 90% power to detect this observed difference?

  

Power and Sample Size

1-Sample t Test

Testing mean = null (versus > null)

Calculating power for mean = null + difference

α = 0.05 Assumed standard deviation = 6.7

Results

Difference	Sample Size	Target Power	Actual Power
3	45	0.9	0.905262

Sample size required = 45.