The manufacturer of the X-15 steel-belted radial truch tire claims that the mean mileage the tire can be driven before the tread wears out is 80,000 km. The standard deviation of the mileage is 8,000 km. The Crosset Truck Company bought 48 tires and found that the mean mileage for their trucks is 79,000 km. From Crosset's experience, can we conclude, at a level of significance of 0.05, that the mean mileage of the tire is less than 80,000?
Z = (mean1 -mean2)/SE (Standard error of the mean)
SE = SD/√(n-1)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion cut off by the Z score found.
To determine whether we can conclude, at a significance level of 0.05, that the mean mileage of the X-15 steel-belted radial truck tire is less than 80,000 km based on Crosset Truck Company's experience, we can perform a hypothesis test. The null hypothesis (H0) assumes that the mean mileage of the tire is 80,000 km, while the alternative hypothesis (Ha) assumes that the mean mileage is less than 80,000 km.
Here are the steps to conduct the hypothesis test:
Step 1: State the hypotheses:
H0: The mean mileage of the tire is 80,000 km.
Ha: The mean mileage of the tire is less than 80,000 km.
Step 2: Set the significance level (α):
In this case, the significance level is given as 0.05, indicating 5% significance.
Step 3: Compute the test statistic:
We can use the z-test for means since the sample size is large (48 tires) and the population standard deviation is known (8,000 km).
The formula for the z-test statistic is:
z = (x̄ - μ) / (σ / √n)
Where:
x̄ = Sample mean (79,000 km)
μ = Hypothesized population mean (80,000 km)
σ = Standard deviation (8,000 km)
n = Sample size (48 tires)
Plugging in the given values, we get:
z = (79,000 - 80,000) / (8,000 / √48)
z = -1,000 / (8,000 / 6.928)
z ≈ -1.62
Step 4: Determine the critical value:
Since the alternative hypothesis is one-tailed (less than), we need to find the critical value for a left-tailed z-test at a significance level of 0.05. Looking up the critical value in a standard normal distribution table or using statistical software, we find it to be approximately -1.645.
Step 5: Compare the test statistic with the critical value:
Since the test statistic (-1.62) is greater than the critical value (-1.645), it falls within the acceptance region. This means we fail to reject the null hypothesis.
Step 6: Make the decision:
Based on the statistical analysis, there is not enough evidence to conclude that the mean mileage of the tire is less than 80,000 km based on Crosset Truck Company's experience at a significance level of 0.05.
In conclusion, the data from Crosset Truck Company does not support the claim that the mean mileage of the X-15 steel-belted radial truck tire is less than 80,000 km.