Question 3 of 6, Step 2 of 3
$6 / 18$
JEFFERY RIPKA
Correct
Gary has discovered a new painting tool to help him in his work. If he can prove to himself that the painting tool reduces the amount of time it takes to paint a room, he has decided to invest in a tool for each of his helpers as well. From records of recent painting jobs that he completed before he got the new tool, Gary collected data for a random sample of 7 medium-sized rooms. He determined that the mean amount of time that it took him to paint each room was 4.4 hours with a standard deviation of 0.3 hours. For a random sample of 6 medium-sized rooms that he painted using the new tool. he found that it took him a mean of 4.2 hours to paint each room with a standard deviation of 0.2 hours. At the 0.02 level, can Gary conclude that his mean time for painting a medium-sized room without using the tool was greater than his mean time when using the tool? Assume that both populations are approximately normal and that the population variances are equal. Let painting times without using the tool be Population 1 and let painting times when using the tool be Population 2.
Step 2 of 3: Compute the value of the test statistic. Round your answer to three decimal places.
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Final Answer: The value of the test statistic, rounded to three decimal places, is \(\boxed{1.431}\).
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 \(\boxed{1.431}\).
