A study of the effect of smoking on sleep patterns is conducted. The measure observed is the time, in minutes, that it takes to fall asleep. These data are obtained:

Smokers: 69.3, 56.0, 22.1, 47.6, 53.2, 48.1, 52.7, 34.4, 60.2, 43.8, 23.2, 13.8

Non-Smokers: 28.6, 25.1, 26.4, 34.9, 28.8, 28.4, 38.5, 30.2, 30.6, 31.8, 41.6, 21.1, 36.0, 37.9, 13.9

Find the variance for both of the given groups.

Which type of variance should I use to find the variance of the above ungrouped data and why?

For the given study on smoking and sleep patterns, the sample variance should be calculated because the data sets are likely samples from larger populations.

Explanation:

When determining the type of variance to calculate for a given set of ungrouped data, it is essential to understand the nature of the data set. If the data set represents the entire population, then you should calculate the population variance. However, if the data set is a sample from a larger population, which is often the case in practical scenarios, you should calculate the sample variance.

In the context of a study comparing the sleep patterns of smokers versus non-smokers, the data sets for both groups are likely samples of larger populations. Therefore, sample variance is more appropriate.

To calculate the sample variance, you would take the sum of the squared differences between each data point and the mean, divided by the number of observations minus one. The minus one accounts for the degrees of freedom in a sample and provides an unbiased estimator of the population variance.