To determine whether there is enough evidence to support the company's claim that the new project management software has reduced the average project completion time, we can perform a hypothesis test for the population mean. Specifically, we will use a one-sample z-test because the population standard deviation is known (15 days), and the sample size (64) is large enough (n ≥ 30).

Step 1: Define the Hypotheses

Let $\mu_0$ be the mean project completion time before the software was implemented.

• Null hypothesis ($H_0$): The new software has not reduced the average project completion time. $H_0: \mu \geq \mu_0 = 85 \text{ days} \ (\text{before software implementation})$

• Alternative hypothesis ($H_a$): The new software has reduced the average project completion time. $H_a: \mu < 85 \text{ days} \ (\text{after software implementation})$

This is a left-tailed test, since the company claims a reduction in project completion time.

Step 2: Calculate the Test Statistic

The test statistic for a one-sample z-test is given by: $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$ Where:

• $\bar{x} = 82 \text{ days}$ is the sample mean,
• $\mu_0 = 85 \text{ days}$ is the population mean before the new software,
• $\sigma = 15 \text{ days}$ is the population standard deviation,
• $n = 64$ is the sample size.

Let’s compute the z-value.

$z = \frac{82 - 85}{\frac{15}{\sqrt{64}}} = \frac{-3}{\frac{15}{8}} = \frac{-3}{1.875} = -1.6$

Step 3: Determine the Critical Value and p-value

Since we are conducting a left-tailed test at a significance level of $\alpha = 0.05$, we need to find the critical z-value that corresponds to $\alpha = 0.05$. From the z-table:

• The critical z-value for a left-tailed test at $\alpha = 0.05$ is approximately -1.645.

Step 4: Make a Decision

• If the test statistic $z$ is less than the critical value, we reject the null hypothesis.

In this case: $z = -1.6 \quad \text{and} \quad z_{\text{critical}} = -1.645$

Since $-1.6$ is greater than $-1.645$, we fail to reject the null hypothesis.

Step 5: Conclusion

At the 5% significance level, there is not enough evidence to support the company's claim that the new project management software has reduced the average project completion time.

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Would you like further details on any of the steps or calculations?

Here are 5 related questions you may find useful:

1. How would the results change if we had a larger sample size?
2. What would happen if the significance level was reduced to 1%?
3. How do we interpret a p-value in hypothesis testing?
4. What assumptions must be met for the z-test to be valid in this scenario?
5. Can we use a t-test instead of a z-test? If so, when?

Tip: Always ensure your sample size is sufficiently large when using z-tests for means to avoid inaccuracies when the sample standard deviation significantly deviates from the population value.