a repair shop believes that people travel more than 3500 miles between oil changes a random sample of 8 cars getting an oil change has average distance of 3375 standard deviation of 225 miles with alpha 0.05 is there enough evidence to support the shops claim
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Is it ≤ .05?
To determine if there is enough evidence to support the repair shop's claim, we can perform a hypothesis test.
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H0): The average distance between oil changes is 3500 miles.
- Alternative Hypothesis (H1): The average distance between oil changes is less than 3500 miles.
Step 2: Set the significance level (alpha).
The significance level (alpha) is given as 0.05.
Step 3: Compute the test statistic.
Given that the sample size (n) is 8, the sample mean (x̄) is 3375 miles, and the sample standard deviation (s) is 225 miles, we can use a one-sample t-test to compute the test statistic value.
t = (x̄ - μ) / (s / √n)
t = (3375 - 3500) / (225 / √8)
t = -125 / (225 / 2.83)
t ≈ -2.78
Step 4: Determine the critical value.
To determine the critical value, we refer to the t-distribution table or use statistical software. Since the alternative hypothesis states that the average distance is less than 3500 miles, we will use a one-tailed test.
With a significance level of 0.05 and degrees of freedom (df) of 7 (n - 1), the critical value is approximately -1.895.
Step 5: Compare the test statistic with the critical value.
Since the test statistic (-2.78) is less than the critical value (-1.895), we can reject the null hypothesis.
Step 6: Interpret the results.
Based on the given data, there is sufficient evidence to support the claim that people travel less than 3500 miles between oil changes.
To determine whether there is enough evidence to support the shop's claim, we can conduct a hypothesis test. Let's break down the steps to perform the hypothesis test:
Step 1: State the Null Hypothesis (H0) and Alternative Hypothesis (Ha):
The null hypothesis assumes that there is no significant difference from the shop's claim, while the alternative hypothesis assumes that there is a significant difference.
H0: The average distance between oil changes is not greater than 3500 miles.
Ha: The average distance between oil changes is greater than 3500 miles.
Step 2: Set the Significance Level (alpha):
The given significance level in this case is alpha = 0.05. This value indicates that there is a 5% chance of committing a Type I error by rejecting the null hypothesis when it is true.
Step 3: Calculate the Test Statistic:
Since we have a sample with a known standard deviation and the sample size (n) is small (n < 30), we will use the t-distribution for the calculation of the test statistic. The formula to calculate the t-test statistic for one-sample is:
t = (sample mean - hypothesized mean) / (sample standard deviation / √n)
In this case:
Sample Mean (x̄) = 3375
Hypothesized Mean (μ0) = 3500
Sample Standard Deviation (s) = 225
Sample Size (n) = 8
Calculating the t-test statistic:
t = (3375 - 3500) / (225 / √8)
Step 4: Determine the Critical Value(s):
The critical value(s) depend on the desired level of significance and the degrees of freedom. Since we have a one-tailed test (claiming that the average distance is greater than 3500 miles), we need to find the critical value that corresponds to the alpha level and the degrees of freedom (n - 1).
With alpha = 0.05 and 7 degrees of freedom (8 - 1), the critical value can be found using a t-distribution table or a statistical software. In this case, the critical value is approximately 1.894.
Step 5: Make a Decision:
If the absolute value of the calculated t-test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, if |t| > 1.894, we reject H0.
Step 6: Calculate the p-value:
The p-value represents the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true.
Using the t-distribution table or a statistical software, we find the p-value associated with the calculated t-test statistic. If the p-value is less than alpha (0.05), we reject the null hypothesis.
Step 7: Analyze Results:
If the p-value is less than alpha, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Follow these steps to perform the calculations and make the decision. Using the provided data and formulas, you should be able to determine whether there is enough evidence to support the shop's claim.