Question 1168294: 1. A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50, 000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Would it be reasonable for the manufacturer to conclude that after 50, 000 miles the population mean amount of tread remaining is 0.30 inches? Use α = 0.05
I need to find Ho, Ha, df, a, p value, decision and conclusion
Please help me thank you :(
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to conduct the hypothesis test to determine if the manufacturer's conclusion is reasonable:
**1. State the Null and Alternative Hypotheses:**
* **Null Hypothesis (H₀):** The population mean amount of tread remaining after 50,000 miles is 0.30 inches.
* H₀: μ = 0.30
* **Alternative Hypothesis (Ha):** The population mean amount of tread remaining after 50,000 miles is different from 0.30 inches.
* Ha: μ ≠ 0.30 (This is a two-tailed test because we are interested in whether the mean is *different* from 0.30, not specifically greater or less than).
**2. Determine the Significance Level (α):**
* The significance level is given as α = 0.05. This means there is a 5% risk of rejecting the null hypothesis when it is actually true.
**3. Calculate the Degrees of Freedom (df):**
* The sample size (n) is 10.
* The degrees of freedom for a t-test with a single sample mean are:
* df = n - 1 = 10 - 1 = 9
**4. Calculate the Test Statistic (t-value):**
* Since the population standard deviation is unknown, we will use a t-test. The formula for the t-statistic is:
* t = (X̄ - μ₀) / (s / √n)
Where:
* X̄ = sample mean = 0.32 inch
* μ₀ = hypothesized population mean = 0.30 inch
* s = sample standard deviation = 0.09 inch
* n = sample size = 10
* t = (0.32 - 0.30) / (0.09 / √10)
* t = 0.02 / (0.09 / 3.162)
* t = 0.02 / 0.02846
* t ≈ 0.703
**5. Determine the P-value:**
* The p-value is the probability of obtaining a sample mean as extreme as (or more extreme than) the one observed, *assuming the null hypothesis is true*.
* Since this is a two-tailed test with df = 9 and a t-value of approximately 0.703, we need to find the area in both tails of the t-distribution beyond t = 0.703 and t = -0.703.
* Looking up a t-distribution table or using a statistical calculator, the p-value for a two-tailed test with t = 0.703 and df = 9 is greater than 0.10 (typically around 0.50).
**6. Make a Decision:**
* We compare the p-value to the significance level (α = 0.05).
* **Decision Rule:**
* If p-value ≤ α, reject the null hypothesis (H₀).
* If p-value > α, fail to reject the null hypothesis (H₀).
* In this case, the p-value (≈ 0.50) is greater than α (0.05). Therefore, we fail to reject the null hypothesis.
**7. State the Conclusion:**
* Based on the sample data and the hypothesis test at a significance level of 0.05, there is not enough statistical evidence to conclude that the population mean amount of tread remaining after 50,000 miles is different from 0.30 inches.
**Therefore, it would be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches.**
**Summary:**
* **H₀:** μ = 0.30
* **Ha:** μ ≠ 0.30
* **df:** 9
* **α:** 0.05
* **t-value:** ≈ 0.703
* **p-value:** > 0.10 (approximately 0.50)
* **Decision:** Fail to reject H₀.
* **Conclusion:** There is not enough statistical evidence to conclude that the population mean amount of tread remaining after 50,000 miles is different from 0.30 inches. It is reasonable for the manufacturer to conclude that the population mean is 0.30 inches.
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