Gaming and distracted eating: A group of researchers are interested in the possible effects of distracting stimuli during eating, such as an increase or decrease in the amount of food consumption. To test this hypothesis, they monitored food intake for a group of 44 patients who were randomized into two equal groups. The treatment group ate lunch while playing solitaire, and the control group ate lunch without any added distractions. Patients in the treatment group ate an average of 52.1 grams of biscuits, with a standard deviation of 45.1 grams, and patients in the control group ate an average of 27.1 grams of biscuits, with a standard deviation of 26.4 grams. Do these data provide convincing evidence that the average food intake (measured in grams of biscuits consumed) is different for the patients in the treatment group? Assume that conditions for inference are satisfied.

Let ${\mu }_{\mathrm{no} \mathrm{distraction}}$ denote the control group ate lunch without any added distraction, and ${\mu }_{\mathrm{distraction}}$ denote the treatment group ate lunch while playing solitaire. The test is that the average food intake is different for the patients in the treatment group. The hypothesis is,

Null hypothesis:

$H_0:{\mu }_{\mathrm{no} \mathrm{distraction}}={\mu }_{\mathrm{distraction}}$

Alternative hypothesis:

$H_a:{\mu }_{\mathrm{no} \mathrm{distraction}}\ne {\mu }_{\mathrm{distraction}}$

The pooled variance is,

$\begin{array}{rcl}{s}_p^2& =& \frac{\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2}{n_1+n_2-2}\\ & =& \frac{\left(22-1\right)26.4^2+\left(22-1\right)45.1^2}{22+22-2}\\ & =& \frac{57350.37}{42}\\ & =& 1,365.485\end{array}$

The pooled variance is 1,365.485.