There has been a lot of debate over whether the flu vaccine works for old people

Question

There has been a lot of debate over whether the flu vaccine works for old people. All we really know is that 1% of old people die during the 1-2 month flu season. How many of them die because of the flu and how many die because of something else is hard to say.

To find out what percent of deaths we can save by the flu shot during flu season, we decide to do a Randomized Controlled Double Blind study. We randomly assign half the subjects to treatment and half the subjects to control. The treatment group gets the real vaccine and the control group gets the placebo before the flu season starts. Then we’ll compare the death rate of the 2 groups at the end of flu season.
We need to plan how many subjects to assign to each group. As we know that depends on the specific alternative hypothesis we choose and the power of the test. Let’s say we're interested in lowering the death rate from 1% to 0.95%. That might sound like a tiny drop but since the elderly population is 46 million, that means 23,000 lives saved!

We can use our usual relation: |Ha-H0|=(|Z|+|Z|)SEdifference

a. As usual, we’ll set the null significance level at 5% and power=80% (since most granting agencies won't fund lower powered studies). First calculate |Z| and |Z|

|Z| = _____

|Z| = _____

b. The null hypothesis is no difference between the 2 groups, the death rate remains at 1%. The alternative hypothesis is that the death rate drops down to 0.95%

|Ha-H0| = _____%

c. Now calculate SEdifference (round to 2 decimal places).

SEdifference = _____%

d. Now that we know SEdifference, we can just calculate SD and then solve for n, the size of each group. (See page 12 if you forget the SEdifference formula.) The null hypothesis is that there’s no real difference between the death rates between the vaccine and the placebo group. That means that they both should have a death rate = 1%.

(i) So the SD of each would be _____

(ii) Now solve for n using the SEdifference n = _____

e. That’s the number of subject for each group, so the total number of subjects in your experiment would have to be

Explanation / Answer

a) Z_alpha for alpha = 0.05 level of significance is 1.64.

Power = 1 - beta, so, beta = 1 - 0.8 = 0.2

Z_beta for beta = 0.2 is 2.05

b) Ha = 0.95% and H0 = 1%

|Ha-H0| = 0.05%

c) SEdifference = (0.05/3.69)% = 0.01%

d) Now, SD = 1% and SDdifference = SD/sqrt(n)

therefore, n = (SD/SDdiff)^2 = 100^2 = 10000

e) Therefore, total number of subject would be 2*10000 = 20000

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