Hypothesis Test (Z-test)

Chen Hao posted on 12 Mar 2016

This is my course notes for Mathematical Biostatistics Boot Camp 2, course materials are free on github.

1. Hypothesis Testing

Hypothesis testing is concerned with making decisions using data

A null hypothesis is specified that represents the status quo, usually labeled H0
The null hypothesis is assumed true and statistical evidence is required to reject it in favor of a research or alternative hypothesis

For example:

2. General rule

3. Two sided tests

4. Confidence intervals

5. P-values

Criticisms of the P-value

6. R test codes

qnorm(.95)

xval <- seq(-3.2, 3.2, length = 1000)
yval<- dnorm(xval)

plot(xval, yval, type = "l", axes = TRUE, frame = FALSE, lwd = 3, xlab = "", ylab = "")
x <- seq(qnorm(.95), 3.2, length = 100)
polygon(c(x, rev(x)),c(dnorm(x), rep(0, length(x))), col = "salmon")
text(mean(x), mean(dnorm(x))+.02, "5%", cex = 2)
text(qnorm(.95), .01, "1.645", cex = 2)

plot(xval, yval, type = "l", axes = TRUE, frame = FALSE, lwd = 3, xlab = "", ylab = "")
x <- seq(qnorm(.975), 3.2, length = 100)
polygon(c(x, rev(x)),c(dnorm(x), rep(0, length(x))), col = "salmon")
text(mean(x), mean(dnorm(x))+.02, "2.5%", cex = 2)
text(qnorm(.975), .01, "1.96", cex = 2)
x <- seq(-3.2, qnorm(.025),length = 100)
polygon(c(x, rev(x)),c(dnorm(x), rep(0, length(x))), col = "salmon")
text(mean(x), mean(dnorm(x))+.02, "2.5%", cex = 2)
text(qnorm(.025), .01, "-1.96", cex = 2)
text(0, dnorm(0) / 5, "95%", cex = 2)

xval <- seq(-4, 4, length = 1000)
yval<- dt(xval, 15)
plot(xval, yval, type = "l", axes = TRUE, frame = FALSE, lwd = 3, xlab = "", ylab = "")
x <- seq(qt(.975, 15), 4, length = 100)
polygon(c(x, rev(x)),c(dt(x, 15), rep(0, length(x))), col = "salmon")
text(mean(x), mean(dt(x, 15))+.02, "2.5%", cex = 2)
text(qt(.975, 15), .01, "2.13", cex = 2)
x <- seq(-4, qt(.025, 15),length = 100)
polygon(c(x, rev(x)),c(dt(x, 15), rep(0, length(x))), col = "salmon")
text(mean(x), mean(dt(x, 15))+.02, "2.5%", cex = 2)
text(qt(.025, 15), .01, "-2.13", cex = 2)
text(0, dt(0, 15) / 5, "95%", cex = 2)

pt(.8, 15, lower.tail=FALSE)
xval <- seq(-4, 4, length = 1000)
yval<- dt(xval, 15)
plot(xval, yval, type = "l", axes = TRUE, frame = FALSE, lwd = 3, xlab = "", ylab = "")
x <- seq(.8, 4, length = 100)
polygon(c(x, rev(x)),c(dt(x, 15), rep(0, length(x))), col = "salmon")
text(mean(x), mean(dt(x, 15))+.02, "22%", cex = 2)
text(0.8, .01, "0.8", cex = 2)