1. The 1-week mortality rate among adult TBI patients is 30%. You conducted a sm

Question

1. The 1-week mortality rate among adult TBI patients is 30%. You conducted a small pilot study among 12 children who had TBI. Six of them died within a week.

(a) You want to test whether the 1-week mortality rate among your study population is the same as the 1-week mortality rate for adult TBI patients. Write down the formula to derive the exact p-value.

(b) Suppose that you want to design a new study with 85% power to test the same hypothesis, based on the estimates from your pilot study. Calculate the necessary sample size by hand.

Explanation / Answer

Solution:

Part a

Here, we have to use the z test for population proportion. The null and alternative hypothesis is given as below:

Null hypothesis: H0: p = 0.3

Alternative hypothesis: Ha: p 0.3

We are given, sample size = n = 12 and x = 6

Sample proportion = P = x/n = 6/12 = 0.5

The test statistic formula is given as below:

Z = (P – p) / sqrt(pq/n)

Where, q = 1 – p = 1 – 0.3 = 0.7 and n = 12

Now, plug all values in the above formula

Z = (0.5 – 0.3) / sqrt(0.3*0.7/12)

Z = 0.2/ 0.132288

Z = 1.511858

P-value = 0.1306 (by using z-table)

Part b

Here, we have to find sample size for the given population proportion.

Sample size formula is given as below:

Sample size = n = pq(Z/E)^2

Where, Z is the critical value at 85% power or confidence level

E is the margin of error which is given as below:

E = Z*sqrt(pq/n)

Z = 1.4395

E = 1.4395* sqrt(0.3*0.7/12)

E = 1.4395*0.132288

E = 0.190429

Sample size = n = 0.3*0.7*(1.4395/0.190429)^2

Sample size = n = 11.99987

Required sample size = 12 approximately

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