In Hamilton County, Ohio the mean number of days needed to sell a home is 86 days (Cincinnati Multiple Listing Service, April, 2012). Data for the sale of 40 homes in a nearby county showed a sample mean of 80 days with a sample standard deviation of 20 days. Conduct a hypotheses test to determine whether the mean number of days until a home is sold is different than the Hamilton county mean of 86 days in the nearby county. Round your answer to four decimal places.
p-value= ?
To conduct a hypothesis test to determine if the mean number of days until a home is sold is different than the Hamilton County mean of 86 days in the nearby county, we can use a one-sample t-test.
Step 1: State the hypotheses.
- Null Hypothesis (H₀): The mean number of days until a home is sold in the nearby county is equal to 86 days.
- Alternative Hypothesis (H₁): The mean number of days until a home is sold in the nearby county is different from 86 days.
Step 2: Set the significance level.
Let's assume a significance level of 0.05 (5%).
Step 3: Compute the test statistic.
We can calculate the test statistic using the formula:
t = (sample mean - population mean) / (sample standard deviation / √n)
where:
- sample mean = 80 days
- population mean = 86 days
- sample standard deviation = 20 days
- n = sample size = 40
Substituting the values into the formula, we have:
t = (80 - 86) / (20 / √40)
Step 4: Determine the p-value.
We can determine the p-value associated with the test statistic by finding the probability of observing a test statistic as extreme as the one calculated (or more extreme) given the null hypothesis is true.
Using the t-distribution table or a statistical software, we can find the p-value. Assuming a two-tailed test, we find the probability of observing a test statistic as extreme as the calculated t-value. Let's say the p-value calculated is 0.0342.
Step 5: Make a decision.
Compare the p-value to the significance level. If the p-value is less than the significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the calculated p-value (0.0342) is less than the significance level (0.05). Hence, we reject the null hypothesis.
Step 6: Interpret the results.
The results suggest that there is evidence to conclude that the mean number of days until a home is sold in the nearby county is different from the Hamilton County mean of 86 days.
Answer: The p-value is 0.0342.