Hypothesis Tests for a Mean

You may have noticed that although we have created confidence intervals for a mean, we have not done hypothesis tests for a mean – until now.

Remember that a randomization distribution should simulate a situation that

• reflects the design of our study and our sample data, and
• forces the null hypothesis to be true.

The randomization distributions we have seen so far have either used

• rflip() to simulate categorical data with a specified proportion, or
• shuffle() to ensure that there is no association between two variables.

Neither of these will work for a hypothesis about a mean. We need a new idea. Here it is:

We can create a randomization distribution for a test about a single mean by shifting the bootstrap distribution to make it centered in the right location.

This will give us a distribution that is centered at the value specified by the null hypothesis and the appropriate amount of variability.

Here’s how.

mean(~BodyTemp, data = BodyTemp50)

## [1] 98.26

# Create the Bootstrap distribution
Temp_boot <- 
  do(2000) * df_stats( ~ BodyTemp, data = resample(BodyTemp50), mean_BodyTemp = mean)
# Shift it so that it is centered in the correct place for our hypothesis
Temp_null <- 
  Temp_boot %>% mutate(mean_BodyTemp = mean_BodyTemp + (98.6 - 98.26))

Notice that 98.6 lies a bit out side the 95% confidence interval and the p-value is a bit less than 0.05.

# 95% confidence interval
cdata(~ mean_BodyTemp, data = Temp_boot)

##      lower upper central.p
## 2.5% 98.05 98.47      0.95

# p-value for test of H0: mu = 98.6
2 * prop(~ (mean_BodyTemp <= 98.26), data = Temp_null)

## prop_TRUE 
##         0