t-Tests

Biology: Cambridge International A Level, topic 17.1.3

Hypothesis Testing

A hypothesis is an assumption made about the value of a particular parameter. We can use a $t$ -test to determine whether the difference between the means of two populations are statistically significant.

Null and alternative hypotheses

The null hypothesis, $H_0$ , is a hypothesis that is assumed to be correct. In a $t$ -test, the null hypothesis is that there is no significant difference between the two population means.
The alternative hypothesis, $H_1$ is that there is a significant difference between the two population means.
A decision is then made as to whether or not there is sufficient evidence to reject the null hypothesis.

$p$ -value

In order to test a hypothesis, we consider how likely the hypothesis is to be true.
The probability of the null hypothesis being true is referred to as the $p$ -value.

Significance level

If the probability of the null hypothesis being true is less likely than a given "significance level", we reject the null hypothesis.
The significance level used in biology is often 5%.

Test statistic

The test statistic used to calculate $p$ -values is equal to:
$t = \frac{\large\overline{x}_1-\overline{x}_2}{\large \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$
$\overline{x}_1$ = is the mean of sample one.
$\overline{x}_2$ = is the mean of sample two.
$s_1$ is the standard deviation of sample one.
$s_2$ is the standard deviation of sample two.
$n_1$ is the number of measures in sample one.
$n_1$ is the number of measures in sample two.

Degrees of freedom

The degrees of freedom is equal to $n_1+n_2-2$ .

Determine the critical value

Once you have determined the value of the test statistic, you need to use a statistics table to determine the critical value.
The critical value will be the value at the significance level (5%) of your particular degrees of freedom.
If the calculated value is greater than the critical value, you reject the null hypothesis.
If the calculated value is less than the critical value, you accept the null hypothesis.

Test yourself on t-Tests

Which symbol denotes the alternative hypothesis?
What is the probability of the null hypothesis being true called?

Practise this topic for free