In 2018 , a study was conducted that stated that the average commute time to work for Kern County residents was 22.6 minutes. Suppose you were hired to determine if the average commute time is different than it was in 2018. You randomly sample 87 individuals who commute to work from Kern county and you find that their average commute time was 23.7 minutes and you find the sample standard deviation of commute times for those 87 individuals was 3.6 minutes. Carry out the appropriate hypothesis test at the $\alpha=0.05$ level of significance to determine if the average commute time differs from what it was in 2018.
Make sure to address all of the following in your response:
- Include your hypotheses. (You can use the Canvas Math Equation editor by clicking on the $\sqrt{x}$ to include the appropriate symbols and notation)
- Check any and all relevant assumptions and state how you determined whether or not they were satisfied (Be as specific as possible)
- Specify your p-value rounded to 3 decimal places. (For example, if you had a p-value of 0.54321 , then you would type 0.543 )
- Include the decision you are making regarding the null hypothesis and specify why you are making that decision.
- Include a 1-sentence conclusion in context of the given problem.
Answer
\(\boxed{\text{Final Answer: The hypotheses for this test were } H_0: \mu = 22.6 \text{ and } H_a: \mu \neq 22.6. \text{ The calculated t-statistic was approximately 2.85 and the p-value was approximately 0.005. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis. Therefore, we have enough evidence to suggest that the average commute time in Kern County has changed since 2018.}}\)
Step 1 ：The problem is asking us to perform a hypothesis test to determine if the average commute time in Kern County has changed since 2018. The null hypothesis is that the average commute time has not changed, i.e., the population mean is still 22.6 minutes. The alternative hypothesis is that the average commute time has changed, i.e., the population mean is not 22.6 minutes.
Step 2 ：We are given the sample size (n=87), the sample mean (x̄=23.7 minutes), and the sample standard deviation (s=3.6 minutes). We are also given the significance level (α=0.05).
Step 3 ：We can use a t-test to test our hypotheses because we do not know the population standard deviation. The t-statistic is calculated as \((x̄ - μ) / (s / \sqrt{n})\), where μ is the population mean.
Step 4 ：The assumptions for the t-test are that the data is normally distributed and that the samples are independent. Since we are given a large sample size (n>30), we can use the Central Limit Theorem to assume that the sample mean is normally distributed. The question states that we randomly sampled 87 individuals, so we can assume the samples are independent.
Step 5 ：After calculating the t-statistic, we can find the p-value using a t-distribution table or a statistical software. The p-value is the probability of observing a t-statistic as extreme as the one calculated, given that the null hypothesis is true. If the p-value is less than α, we reject the null hypothesis. If the p-value is greater than α, we fail to reject the null hypothesis.
Step 6 ：The calculated t-statistic is approximately 2.85 and the p-value is approximately 0.005. Since the p-value is less than the significance level (0.005 < 0.05), we reject the null hypothesis. This means that we have enough evidence to suggest that the average commute time in Kern County has changed since 2018.
Step 7 ：\(\boxed{\text{Final Answer: The hypotheses for this test were } H_0: \mu = 22.6 \text{ and } H_a: \mu \neq 22.6. \text{ The calculated t-statistic was approximately 2.85 and the p-value was approximately 0.005. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis. Therefore, we have enough evidence to suggest that the average commute time in Kern County has changed since 2018.}}\)
