Question

Dr. Sellers is planning a study to investigate the effect of a low sugar diet on obese postmenopausal women. The goal of the diet is to reduce urinary sucrose (a type of sugar). From previous studies, Dr. Sellers knows that postmenopausal obese women who are not on low sugar diets have a mean urinary sucrose of 25 mg/d with a standard deviation of 18, and Dr. Sellers would like to have adequate power to test whether women on the low sugar diet have urinary sucrose values of 10 mg/d.

a) What sample size will be required if Dr. Sellers would like to have 80% power for a two-sided test and a type 1 error rate of 5%?

b) What would happen to the required sample size in the previous problem if Dr. Sellers wanted to test whether the low sugar diet lowers the urinary sucrose to a value of 5 (instead of 10)? (Note: You do not need to compute the sample size, just give a brief justification for if it would go up or down.)

Answer 1

a)

Null hypothesis H0: $\mu$ = 25 mg/d

Alternative hypothesis H0: $\mu$$\ne$ 25 mg/d

Effect size, ES = $\left | \mu_1 - \mu_0 \right | / \sigma$

= |10 - 25| / 18 = 0.8333

Sample size to ensure that the test has a specified power is

----(1)

rl ES

where $\alpha$ and $\beta$ are type I and II error respectively.

$\alpha$ = 0.05, 1 - $\beta$ = 0.80

For 80% power Z_0.80 = 0.84

Z₀₉₇₅ = 1.96

So,

n = ((1.96 + 0.84) / 0.8333)²

= 11 (Rounding to nearest integer)

b)

For $\mu_1$ = 5 , $\left | \mu_1 - \mu_0 \right |$ will increase and thus ES will increase and consequently, the sample size will decrease (as in formula, n is inversely proportional to ES).