Question 4 of 5 , Step 2 of 3
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A car company claims that its new SUV gets better gas mileage than its competitor's SUV. A random sample of 41 of its sUVs has a mean gas mileage of 13.8 miles per gallon (mpg). The population standard deviation is known to be $1.4 \mathrm{mpg}$. A random sample of 32 competitor's suvs has a mean gas mileage of $13.4 \mathrm{mpg}$. The population standard deviation for the competitor is known to be $0.3 \mathrm{mpg}$. Test the company's claim at the 0.10 level of significance. Let the car company's suvs be Population 1 and let the competitor's SuVs be Population 2.

Step 2 of 3 : Compute the value of the test statistic. Round your answer to two decimal places.
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Step 1 ：Given that the car company's SUVs are Population 1 and the competitor's SUVs are Population 2, we have the following data:

Step 2 ：Population 1: Sample size (n1) = 41, Sample mean (X1) = 13.8 mpg, Population standard deviation (σ1) = 1.4 mpg

Step 3 ：Population 2: Sample size (n2) = 32, Sample mean (X2) = 13.4 mpg, Population standard deviation (σ2) = 0.3 mpg

Step 4 ：We are testing the company's claim at the 0.10 level of significance.

Step 5 ：The test statistic for a hypothesis test comparing two means can be calculated using the formula: \(Z = \frac{(X1 - X2)}{\sqrt{(\frac{σ1^2}{n1}) + (\frac{σ2^2}{n2})}}\)

Step 6 ：Substituting the given values into the formula, we get: \(Z = \frac{(13.8 - 13.4)}{\sqrt{(\frac{1.4^2}{41}) + (\frac{0.3^2}{32})}}\)

Step 7 ：Calculating the above expression, we get \(Z = 1.7779116213900734\)

Step 8 ：Rounding the test statistic to two decimal places, we get \(Z = 1.78\)

Step 9 ：Final Answer: The value of the test statistic, rounded to two decimal places, is \(\boxed{1.78}\)