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.