a) H₀: µ = 12 (mean call duration hasn't changed) vs. Ha: µ > 12 (mean call duration has increased)
b) Test statistic (z) = (12.8 - 12) / (1.9 / √41) ≈ 1.74
c) Critical value (z_α) for a one-tailed test at α = 0.05 is ≈ 1.645
d) Calculated z > critical z, so we reject the null hypothesis.
e) p-value ≈ 0.04 ≈ α, so we reject the null hypothesis.
f) Based on the rejection of the null hypothesis, there is evidence to suggest that the average call duration has increased significantly after the changes in call handling procedures.
a) Null and Alternative Hypotheses:
H₀ (Null Hypothesis): The mean call duration remains unchanged at 12 minutes (µ = 12).
Ha (Alternative Hypothesis): The mean call duration has increased significantly after the changes (µ > 12).
We assume H₀ to be true until compelling evidence suggests otherwise.
b) Calculated Test Statistic:
The z-statistic for one-tailed test with unequal variance is used:
z = (Sample Mean - Population Mean) / (Sample Standard Deviation / √Sample Size)
Plugging the values:
z = (12.8 - 12) / (1.9 / √41) ≈ 1.74
c) Critical Value of the Test Statistic:
At a significance level of α = 0.05 and a one-tailed test (Ha is directional), the critical z-value is approximately 1.645.
d) Decision about the Null Hypothesis:
Since the calculated z-value (1.74) is greater than the critical z-value (1.645), we reject the null hypothesis (H₀).
e) p-value and Comparison:
The p-value, the probability of observing a statistic as extreme as or more extreme than the calculated one when H₀ is true, is roughly 0.04. As it's less than the significance level (α = 0.05), we again reject the null hypothesis.
f) Conclusion: Based on both the z-test and the p-value, there is statistically significant evidence (at the 0.05 level) to reject the null hypothesis and conclude that the average call duration at QuickHelp has indeed increased after the changes in call handling procedures.