To see whether people are keeping their car tires inflated to the correct level of 35 pounds per square inch (psi), a tire company manager selects a sample of 36 tires and checks the pressure. The mean of the sample is 33.5 psi, and the population standard deviation is 3 psi. Are the tires properly inflated? Use α = 0.10.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to your Z score.
To determine whether the tires are properly inflated, we need to conduct a hypothesis test using the given information. Here are the steps to follow:
Step 1: State the null hypothesis (H₀) and the alternative hypothesis (H₁):
- Null hypothesis (H₀): The mean tire pressure is equal to 35 psi (µ = 35).
- Alternative hypothesis (H₁): The mean tire pressure is not equal to 35 psi (µ ≠ 35).
Step 2: Set the significance level (α) to 0.10. This is the probability of rejecting the null hypothesis when it is true.
Step 3: Calculate the test statistic. Since we know the population standard deviation (σ), we can use the z-test formula:
- z = (x̄ - µ) / (σ / √n)
where:
- x̄ is the sample mean (33.5 psi),
- µ is the hypothesized population mean (35 psi),
- σ is the population standard deviation (3 psi), and
- n is the sample size (36 tires).
Plugging in the values, we get:
- z = (33.5 - 35) / (3 / √36)
Simplifying, we have:
- z = (33.5 - 35) / (3 / 6)
- z = -1.5 / 0.5
- z = -3
Step 4: Determine the critical value(s) based on the significance level (α). Since α = 0.10 and it is a two-tailed test, we need to find the critical z-scores that enclose 0.10 probability in both tails.
Using a standard normal distribution table or calculator, we find the critical z-scores to be approximately -1.645 and 1.645. These values correspond to a cumulative probability of 0.05 in each tail.
Step 5: Compare the test statistic (z) with the critical value(s).
- If the test statistic falls within the critical values, we fail to reject the null hypothesis.
- If the test statistic falls outside the critical values, we reject the null hypothesis.
In this case, since -3 falls outside the critical values of -1.645 and 1.645, we reject the null hypothesis.
Step 6: Interpretation:
Since we rejected the null hypothesis, we can conclude that the mean tire pressure is significantly different from 35 psi. Therefore, based on the sample data, the tires are not properly inflated at the correct level of 35 psi.
Note: Remember to always refer to the appropriate critical values and consider whether it's a one-tailed or two-tailed test based on the nature of the alternative hypothesis.