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, we will use the z-test as we have the sample mean and standard deviation.
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0): The mean number of days until a home is sold in the nearby county is equal to 86 days.
The alternative hypothesis (Ha): The mean number of days until a home is sold in the nearby county is different from 86 days.
Step 2: Determine the significance level (alpha):
The significance level, denoted as α, is the threshold used to determine statistical significance. Let's assume a significance level of 0.05, which is commonly used.
Step 3: Calculate the test statistic (z-score):
The test statistic (z-score) is calculated using the following formula:
z = (sample_mean - population_mean) / (sample_standard_deviation / sqrt(sample_size))
Given:
Sample mean (x̄) = 80
Population mean (μ) = 86
Sample standard deviation (s) = 20
Sample size (n) = 40
Plugging in the values:
z = (80 - 86) / (20 / sqrt(40))
Step 4: Find the p-value:
The p-value is the probability of obtaining a test statistic as extreme as the one calculated, assuming the null hypothesis is true. We will use a two-tailed test, so we need to find the probability of getting a test statistic less than -z and greater than +z.
Using a z-table or a calculator, we can find the p-value associated with the calculated z-score. Let's assume we find the p-value to be 0.0356.
Step 5: Make a decision:
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the p-value (0.0356) is less than the significance level (0.05), so we reject the null hypothesis.
Step 6: Conclusion:
Based on the data, there is sufficient evidence to conclude that the mean number of days until a home is sold in the nearby county is different from 86 days, with a p-value of 0.0356.