The Sledge Tire and Rubber Company plans to warranty its new mountain bike tire for 12 months. However, before it does this, the company wants to be sure that the mean lifetime of the tires is at least 18 months under normal operations. It will put the warranty in place unless the sample data strongly suggest that the lifetime of the tires is less than 18 months. The company plans to test this statistically using a random sample of tires. The test will be conducted using an alpha level of 0.03.
To test whether the mean lifetime of the mountain bike tires is at least 18 months, the Sledge Tire and Rubber Company will conduct a hypothesis test. The company plans to use a random sample of tires and a significance level, or alpha level, of 0.03.
Here's the step-by-step explanation on how the company can conduct this hypothesis test:
Step 1: Formulate the Null and Alternative Hypotheses
The first step is to state the null and alternative hypotheses. In this case, the null hypothesis (H0) would be that the mean lifetime of the tires is indeed 18 months or longer. The alternative hypothesis (Ha) would be that the mean lifetime of the tires is less than 18 months.
H0: μ ≥ 18
Ha: μ < 18
Step 2: Determine the Test Statistic
The company needs to choose an appropriate test statistic to analyze the sample data. Since the population standard deviation is not provided, and the sample size is likely to be small, it is appropriate to use the t-test statistic.
Step 3: Choose the Significance Level
In this case, the significance level, or alpha level, is already given as 0.03. This value represents the maximum probability of rejecting the null hypothesis when it is true.
Step 4: Conduct the Test and Calculate the P-value
The company will collect a random sample of tires and measure their lifetimes. Based on the sample data, they will calculate the test statistic and find the corresponding p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
Step 5: Make a Decision
Finally, the company will compare the p-value to the significance level (alpha) to make a decision. If the p-value is less than alpha, the company will reject the null hypothesis and conclude that the lifetime of the tires is less than 18 months. If the p-value is greater than or equal to alpha, the company will fail to reject the null hypothesis, indicating that the lifetime of the tires is equal to or greater than 18 months.