Quartz 4

Sampling Distribution of the Difference of Proportions

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Definition

It is a specific type of Sampling Distribution

Let 𝑋 be a probability function 𝑓(𝑥) and 𝑌 the probability function of 𝑓(𝑦), then same as with the Sampling Distribution of a Proportion we have:

𝐸(̂𝑝𝑋)=̂𝑝𝑋, Var(̂𝑝𝑋)=̂𝑝𝑋(1−̂𝑝𝑋)𝑛1,𝐸(̂𝑝𝑌)=̂𝑝𝑌, Var(̂𝑝𝑌)=̂𝑝𝑌(1−̂𝑝𝑌)𝑛2

Therefore we can get:

𝐸(̂𝑝𝑋−̂𝑝𝑌)=̂𝑝𝑋−̂𝑝𝑌Var(̂𝑝𝑋−̂𝑝𝑌)=̂𝑝𝑋(1−̂𝑝𝑋)𝑛1+̂𝑝𝑌(1−̂𝑝𝑌)𝑛2

Normal Approximation

For a sufficiently large 𝑛 and 𝑚, 𝑋 and 𝑌 are distributted as normal distributions. Then 𝑋−𝑌 also

𝐸(̂𝑝𝑋)−𝐸(̂𝑝𝑌)=𝑁(̂𝑝𝑋−̂𝑝𝑌,√̂𝑝𝑋(1−̂𝑝𝑋)𝑛1+̂𝑝𝑌(1−̂𝑝𝑌)𝑛2→𝑍=(𝑋−𝑌)−(𝜇𝑋−𝜇𝑌)√𝜎2𝑋𝑛+𝜎2𝑌𝑚

Solution Methods

Standard Deviations Known

𝑇=(̄𝑋1−̄𝑋2)−(𝜇1−𝜇2)√𝑆2𝑝⋅(1𝑛1+1𝑛2)𝑆2𝑝=(𝑛1−1)𝑆21+(𝑛2−1)𝑆22𝑛1+𝑛2−2

In R:

pnorm(x, mu1 − mu2, sqrt(sd1^2/n1 + sd2^2/n2))

Standard Deviations Unknown but equal

𝑇(𝑛−1)=𝑋−𝜇𝑆/√𝑛

In R:

df  <- n1 + n2 - 2
Sp  <- sqrt(((n1-1)*S1^2 + (n2-1)*S2^2) / df)
 
t_val <- ((xbar1-xbar2) - (mu1-mu2)) / (Sp*sqrt(1/n1+1/n2))
pt(t_val, df=df)

Standard Deviations Unknown and different

𝑇=(̄𝑋1−̄𝑋2)−(𝜇1−𝜇2)√𝑆21𝑛1+𝑆22𝑛2𝑣=(𝑆21𝑛1+𝑆22𝑛2)2(𝑆21𝑛1)2𝑛1−1+(𝑆22𝑛2)2𝑛2−1

In R:

num_df <- (s1^2/n1 + s2^2/n2)^2
den_df <- ((s1^2/n1)^2 / (n1 - 1)) + ((s2^2/n2)^2 / (s2 - 1))
df <- num_df / den_df
 
diff_observed <- xbar1 - xbar2
mean_theoretical <- mu1 - mu2
stderr <- sqrt(s1^2/n1 + s2^2/n2)
 
t_val <- (diff_observed - mean_theoretical) / stderr
pt(t_val, df = df)

Graph View