A countrywide estate agency specializes in selling commercial ventures. Their records show that the mean selling time is less than 90 days. Because of recent economic conditions, they believe that the mean selling time is now greater than 90 days. A countrywide survey of 100 businesses sold recently revealed that the mean selling time was 94 days, with a standard deviation of 22 days. At the 0.10 lavel of significance, has there been an increase in selling time?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
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.
z= 90-94/22/√100
Determine and interpret Pearson’s correlation coefficient.
To determine if there has been an increase in selling time, we can conduct a hypothesis test.
Let's set up our hypotheses:
Null hypothesis (H0): The mean selling time is equal to 90 days.
Alternative hypothesis (H1): The mean selling time is greater than 90 days.
Next, we need to calculate the test statistic. Since we are working with a sample and know the population standard deviation, we can use the z-test.
The formula for the test statistic (z) is:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
In this case, the sample mean (x̄) is 94 days, the population mean (μ) is 90 days, the population standard deviation (σ) is 22 days, and the sample size (n) is 100.
Plugging these values into the formula, we get:
z = (94 - 90) / (22 / sqrt(100))
z = 4 / (22 / 10)
z = 4 / 2.2
z ≈ 1.82
Now, we need to determine the critical value for a significance level of 0.10. Since we are testing for an increase in selling time, which corresponds to the right-tail of the distribution, the critical value is found using the Z-table or a statistical software. At a significance level of 0.10, the critical value is approximately 1.28.
Finally, we compare the test statistic (1.82) to the critical value (1.28). If the test statistic is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, 1.82 > 1.28, so we can reject the null hypothesis. Therefore, we have evidence to suggest that there has been an increase in selling time at the 0.10 level of significance.