Step 1 :The problem is asking for the value of the test statistic in a hypothesis test comparing the means of two populations. The populations in this case are the times it takes Gary to paint a room without and with the new tool.
Step 2 :The test statistic for a two-sample t-test (assuming equal variances) is given by the formula: \(t = (\bar{x}_1 - \bar{x}_2) / \sqrt{(s_1^2/n_1) + (s_2^2/n_2)}\) where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.
Step 3 :We can plug in the given values into this formula to calculate the test statistic. Here are the given values: \(\bar{x}_1 = 4.4\), \(s_1 = 0.3\), \(n_1 = 7\), \(\bar{x}_2 = 4.2\), \(s_2 = 0.2\), \(n_2 = 6\).
Step 4 :Substituting these values into the formula, we get \(t = 1.4313561708410947\).
Step 5 :Final Answer: The value of the test statistic, rounded to three decimal places, is \(\boxed{1.431}\).