Questions: A car company claims that its new SUV gets better gas mileage than its competitor's SUV. A random sample of 35 of its SUVS has a mean gas mileage of 16.7 miles per gallon (mpg). The population standard deviation is known to be 0.9 mpg. A random sample of 42 competitor's SUVs has a mean gas mileage of 16.4 mpg. The population standard deviation for the competitor is known to be 0.6 mpg. Test the company's claim at the 0.02 level of significance. Let the car company's SUVs be Population 1 and let the competitor's SUVs be Population 2.

A car company claims that its new SUV gets better gas mileage than its competitor's SUV. A random sample of 35 of its SUVS has a mean gas mileage of 16.7 miles per gallon (mpg). The population standard deviation is known to be 0.9 mpg. A random sample of 42 competitor's SUVs has a mean gas mileage of 16.4 mpg. The population standard deviation for the competitor is known to be 0.6 mpg. Test the company's claim at the 0.02 level of significance. Let the car company's SUVs be Population 1 and let the competitor's SUVs be Population 2.
Transcript text: A car company claims that its new SUV gets better gas mileage than its competitor's SUV. A random sample of 35 of its SUVS has a mean gas mileage of 16.7 miles per gallon (mpg). The population standard deviation is known to be 0.9 mpg. A random sample of 42 competitor's SUVs has a mean gas mileage of 16.4 mpg. The population standard deviation for the competitor is known to be 0.6 mpg. Test the company's claim at the 0.02 level of significance. Let the car company's SUVs be Population 1 and let the competitor's SUVs be Population 2.
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Solution

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Step 1: State the null and alternative hypotheses
  • Null Hypothesis \(H_0: \mu_1 = \mu_2\)
  • Alternative Hypothesis \(H_a: \mu_1 > \mu_2\)
Step 2: Calculate the test statistic
  • Test Statistic \(Z\) is calculated using the formula: \[Z = \frac{(16.7 - 16.4)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\]
  • Substituting the values, we get \(Z = 1.685\)
Step 3: Determine the critical value(s)
  • Critical Z-value(s) for a significance level of \(lpha = 0.02\) is/are 2.054
Step 4: Make a decision
  • The test statistic does not fall in the rejection region, thus we fail to reject \(H_0\).
Step 5: Draw a conclusion
  • There is not sufficient evidence to support the claim about the population means.
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