Steve Cutter sells Big Blade lawn mowers in his hardware store, and he is interested in comparing
the reliability of the mowers he sells with the reliability of Big Blade mowers sold nationwide. Steve
knows that only 15 percent of all Big Blade mowers sold nationwide requires repairs during the first
year of ownership. A sample of 120 of Steve’s customers revealed that exactly 22 of them required
mower repairs in the first year of ownership. At a 0.02 level of significance, is there evidence that
Steve’s Big Blade mowers differ in reliability from those sold nationwide?
To determine if there is evidence that Steve's Big Blade mowers differ in reliability from those sold nationwide, we can perform a hypothesis test.
Step 1: State the null and alternative hypothesis
- Null hypothesis (H0): The reliability of Steve's Big Blade mowers is the same as those sold nationwide.
- Alternative hypothesis (Ha): The reliability of Steve's Big Blade mowers is different from those sold nationwide.
Step 2: Choose a significance level (alpha)
- The significance level (alpha) in this case is 0.02, which means we are willing to accept a 2% chance of rejecting the null hypothesis when it is actually true.
Step 3: Collect sample data and calculate the test statistic
- From the information given, we know that Steve's sample of 120 customers had 22 who required mower repairs in the first year of ownership. This gives us a sample proportion of repairs, p̂ = 22/120 = 0.1833.
Step 4: Determine the test statistic
- To determine the test statistic, we need to calculate the z-score using the sample proportion and the population proportion.
- The population proportion, p, is given as 0.15 (15%).
- The formula for the z-score is: z = (p̂ - p) / sqrt((p * (1-p)) / n), where n is the sample size.
- Substituting the values, we get z = (0.1833 - 0.15) / sqrt((0.15 * (1-0.15)) / 120).
Step 5: Determine the critical value(s)
- Since the alternative hypothesis is "different," this is a two-tailed test.
- We need to find the critical values that correspond to a significance level of 0.02/2 = 0.01 in each tail.
- Using a standard normal distribution table or calculator, we find the critical z-values for a two-tailed test are approximately -2.576 and 2.576.
Step 6: Compare the test statistic with the critical value(s)
- If the test statistic falls within the range of the critical values, we fail to reject the null hypothesis. Otherwise, we reject the null hypothesis.
- In this case, if the calculated z-score is less than -2.576 or greater than 2.576, we reject the null hypothesis.
Step 7: Make a decision
- Calculate the z-score using the formula given in Step 4.
- If the calculated z-score falls outside the range of -2.576 to 2.576, we reject the null hypothesis.
- Compare the calculated z-score with the critical values to make a decision.
In summary, to determine if there is evidence that Steve's Big Blade mowers differ in reliability from those sold nationwide, perform the steps outlined above.