# Lesson 11—Two-Sample t tests

#### By Olivia Mah

First published on April 20, 2019

### Learning Objectives

Perform a two-sample t hypothesis test

### Demonstration

Goal:

To conduct a two-sample t test with a two-sided alternative hypothesis. The following example to test if there is a difference between heights of plants grown with and without fertilizers (see p111 in [1]).

Here are the null and alternative hypotheses in this example:

$$H_0: \mu_1 = \mu_2$$

$$H_a: \mu_1 \neq \mu_2$$

where

$$\mu_1$$ = population mean height of plants grown without fertilizers

$$\mu_2$$= population mean height of plants grown with fertilizers

Here are the steps:

#### Step 1: Enter the data

We create a vector called cont to store heights of plants grown without fertilizers.

cont = c(64.7, 86.6, 67.1, 62.6, 75.1, 83.8, 71.7, 83.4, 90.3, 82.7)



We then create another vector called fert to store heights of plants grown with fertilizers.

fert = c(110.3, 130.4, 114.0, 135.7, 129.9, 98.2, 109.4, 131.4, 127.9, 125.7)



#### Step 3: Draw boxplots to check data

We draw two boxplots to check if the data are roughly symmetric and without too many extreme outliers:

boxplot(cont, fert, names =c("Control", "Fertilizer"),
xlab = "Treatment", ylab = "Plant Height (cm)",
main = "Plants with(out) Fertilizer", cex.lab =1.5)


Explanation:

The argument cex.lab magnifies the labels (default value is 1).

#### Step 4: Run the two-sample t test via the R function t.test

t.test(cont, fert, mu = 0, conf.level = 0.99)

Explanation:

Whenever R runs a hypothesis test, R automatically calculates the corresponding confidence interval —the range of values which the population mean is estimated to lie within.

Given a set of data, the corresponding hypothesis test result and the confidence interval are closely related. Therefore if we want the significance level $$\alpha$$ to be 0.01, then we set the argument conf.level = 0.99  because  conf.level = 1 – $$\alpha$$ .

By default, R automatically sets  mu=0  and conf.level = 0.95 even if you don’t explicitly type these arguments. So you can skip typing these arguments into the t.test function if you are testing a two-sided alternative hypothesis with  $$\alpha =0.05$$.

### References

[1] Hartvigsen, G. 2014. A Premier in Biological Data Analysis and Visualization Using R. Columbia University Press.

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