A medical research group is recruiting people to complete short surveys about their medical history. For example, one survey asks for information on a person’s family history in regards to cancer. Another survey asks about what topics were discussed during the person’s last visit to a hospital. So far, as people sign up, they complete an average of just 4 surveys, and the standard deviation of the number of surveys is about 2.2. The research group wants to try a new interface that they think will encourage new enrollees to complete more surveys, where they will randomize each enrollee to either get the new interface or the current interface. How many new enrollees do they need for each interface to detect an effect size of 0.5 surveys per enrollee, if the desired power level is 80%?

\(n = 2\frac{((Z_{\alpha/2} + Z_\beta)^2)(s^2)}{S^2}\)

\(s = 2.2\)

\(S = 0.5\)

\(1 - \beta = .80\)

\(\beta = .20\)

\(\alpha = 0.05\)

imgage <- "C:/Users/jpsim/Documents/Stat & Probability for Data/ztable.png"
include_graphics(imgage)

\(n = 2(\frac{((Z_{0.05/2} + Z_{0.20})^2)(2.2^2)}{0.5^2})\)

\(n = 2\frac{((1.96 + 0.84162)^2)(4.28)}{0.25}\) (from Z-Table)

n <- ((2 * ((1.96+0.84162)*(1.96+0.84162)) * (2.2*2.2))/ 0.25)
n

## [1] 303.9162

S1 <- 2.2
S2 <- 2.2

# Minimum Effect Size
E <- 0.5

# Critical Value when alpha = 0.05
Z1 <- qnorm(0.975)

# Critical Value of 80% Confidence Interval
Z2 <- qnorm(0.80)

# Sample Size Determination:
# Effect Size = (z1+z2) * SE = (z1+z2)*(s1^2/n + s2^2/n)
n2 <- (Z1 +Z2)^2 * (S1^2 + S2^2) / E^2
n2

## [1] 303.9086

Therefore, the number of enrolles they need is 304

n

## [1] 303.9162

n2

## [1] 303.9086