To determine if there is enough evidence to support the claim that the new project management software has reduced the average project completion time, we can conduct a hypothesis test. Specifically, we'll use a one-sample Z-test for the population mean since the population standard deviation is known.

Hypothesis Setup:

1. Null hypothesis $H_0$: The average project completion time is not reduced, i.e., it is still 85 days. $H_0: \mu = 85 \text{ days}$
2. Alternative hypothesis $H_a$: The average project completion time has been reduced, i.e., it is less than 85 days. $H_a: \mu < 85 \text{ days}$

Step 1: Gather the information

• Population standard deviation $\sigma = 15$ days
• Sample size $n = 64$
• Sample mean $\bar{x} = 82$ days
• Population mean $\mu_0 = 85$ days (under the null hypothesis)
• Significance level $\alpha = 0.05$

Step 2: Calculate the test statistic (Z)

The test statistic for a one-sample Z-test is calculated using the formula: $Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Substitute the given values:

= \frac{-3}{\frac{15}{8}} = \frac{-3}{1.875} = -1.6$$ ### Step 3: Determine the critical value For a left-tailed Z-test at a 5% significance level, the critical Z-value (from Z-tables) is: $$Z_{\alpha} = -1.645$$ ### Step 4: Make a decision - If the test statistic $$Z$$ is less than the critical value $$Z_{\alpha} = -1.645$$, we reject the null hypothesis. - In our case, $$Z = -1.6$$ which is **greater** than $$-1.645$$. ### Step 5: Conclusion Since $$Z = -1.6$$ is **not less** than $$Z_{\alpha} = -1.645$$, we **fail to reject the null hypothesis**. This means there is **not enough evidence** at the 5% significance level to support the claim that the new project management software has reduced the average project completion time. --- Would you like further details on any part of this process or have any questions? Here are 5 related questions to deepen understanding: 1. How would the conclusion change if the significance level was set at 10% instead of 5%? 2. What would happen if the sample size was smaller, say 36 projects? 3. How does knowing the population standard deviation impact the choice of test (Z-test vs. t-test)? 4. Can we interpret the p-value in this context and what does it represent? 5. How can the power of the test be increased in this situation? **Tip:** Increasing the sample size generally reduces the variability in your sample mean, which can make it easier to detect small effects or differences.