This is a hypothesis testing problem, where the company claims that the new project management software has reduced the average project completion time. We will perform a one-tailed z-test to test this claim, as we are interested in seeing if the mean project completion time has decreased.

Steps for hypothesis testing:

1. State the hypotheses:

   • Null hypothesis ($H_0$): The mean project completion time has not reduced. $H_0: \mu \geq 85$
   • Alternative hypothesis ($H_a$): The mean project completion time has reduced. $H_a: \mu < 85$ Here, $\mu$ represents the true population mean project completion time, and 85 days is the past average project completion time.

2. Known values:

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

3. Test statistic: Since the population standard deviation is known and the sample size is large ($n > 30$), we can use the z-test formula: $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$ Where:

   • $\bar{x}$ = sample mean = 82
   • $\mu_0$ = population mean under the null hypothesis = 85
   • $\sigma$ = population standard deviation = 15
   • $n$ = sample size = 64

4. Compute the test statistic: Let's calculate the z-value.

$z = \frac{82 - 85}{\frac{15}{\sqrt{64}}}$

We can now compute this value.The computed test statistic (z-value) is $-1.6$.

5. Determine the critical value: Since this is a one-tailed test (lower tail), and the significance level ($\alpha$) is 0.05, we need to find the z-critical value corresponding to a 5% significance level. The critical z-value for a left-tailed test at $\alpha = 0.05$ is approximately $-1.645$.

6. Compare the test statistic with the critical value:

   • The computed z-value is $-1.6$.
   • The critical z-value is $-1.645$.

7. Conclusion: Since the computed z-value ($-1.6$) is greater than the critical z-value ($-1.645$), we fail to reject the null hypothesis.

Final Decision:

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

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Do you need any further details or have questions?

5 Related Questions:

1. What is the difference between one-tailed and two-tailed tests?
2. How is the significance level ($\alpha$) chosen in hypothesis testing?
3. What is the importance of the population standard deviation in z-tests?
4. How would the result change if the sample size were smaller?
5. What is the impact of increasing the significance level?

Tip:

Always check the assumptions for a z-test, such as whether the population standard deviation is known and if the sample size is large enough for the Central Limit Theorem to apply.