• Complete a two-sample [latex]t[/latex]-test for independent population means from hypotheses to conclusions

Let’s visualize the difference in means.

[Trouble viewing? Click to open in a new tab.]

The analysis above uses descriptive statistics only. How can we make an inference about the increase in hate crimes in the United States?

We can make an inference using a hypothesis test to provide evidence that there is an increase in the number of hate crimes between 2019 and 2020. A hypothesis test is needed to go beyond the visualizations and show that the difference is not simply sampling variability.

Standard Error of the Difference of Means

We can use a hypothesis test to determine if the observed difference in sample means is consistent with a hypothesized difference in population means.

To do this, we use what we know about the sampling distribution of [latex]\bar{x}_1-\bar{x}_2[/latex] and, in particular, its estimated standard deviation (the standard error). Recall that you learned that the difference in the sample means, [latex]\bar{x}_1-\bar{x}_2[/latex], also has an approximately normal distribution, centered at the difference of the population means, [latex]\mu_1-\mu_2[/latex].

The standard deviation is given by the following formula: [latex]\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}[/latex]

In practice, we will have to estimate the standard deviation because it depends on the unknown population standard deviations. Replacing [latex]\sigma_1[/latex] and [latex]\sigma_2[/latex] by the sample standard deviations [latex]s_1[/latex] and [latex]s_2[/latex], we will get the standard error of the difference.

standard error of the difference of means

standard error of [latex]\bar{x}_1-\bar{x}_2[/latex]: [latex]\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}[/latex]

Now that we have the standard error, we can calculate the test statistic, P-value, and make an inference about the population. We can leverage technology to generate our statistical output and use it to interpret our results.