A software development company claims that its new project management software has reduced the average time required to complete projects. According to past data, the population standard deviation for project completion time is known to be 15 days. The company wants to test this claim using a 5% significance level. They collect a sample of 64 completed projects and find a sample mean completion time of 82 days. Is there enough evidence to support the company's claim that the new software has reduced the average project completion time?

To determine whether there is enough evidence to support the company's claim that the new software has reduced the average project completion time, we can perform a hypothesis test using the following steps:

### Step 1: State the Hypotheses
- **Null Hypothesis (\(H_0\))**: The average project completion time has not been reduced, meaning the mean completion time is equal to or greater than the previous mean. Assuming the previous mean completion time is \( \mu_0 \).
  \[
  H_0: \mu \geq \mu_0
  \]

- **Alternative Hypothesis (\(H_1\))**: The average project completion time has been reduced, meaning the mean completion time is less than the previous mean.
  \[
  H_1: \mu < \mu_0
  \]

Since the specific previous mean (\(\mu_0\)) is not provided, we will proceed under the assumption that \( \mu_0 \) is the historical average project completion time.

### Step 2: Determine the Test Statistic
Since the population standard deviation (\(\sigma = 15\) days) is known and the sample size is large (\(n = 64\)), we will use the **z-test** for the mean. The test statistic \(z\) is calculated as follows:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{x}\) is the sample mean
- \(\sigma\) is the population standard deviation
- \(n\) is the sample size

Substituting the values:
\[
z = \frac{82 - \mu_0}{\frac{15}{\sqrt{64}}}
\]
\[
z = \frac{82 - \mu_0}{\frac{15}{8}}
\]
\[
z = \frac{82 - \mu_0}{1.875}
\]

### Step 3: Determine the Critical Value and Significance Level
For a one-tailed test at a 5% significance level (\(\alpha = 0.05\)), the critical value \(z_{crit}\) is approximately \(-1.645\) (this value is negative because we are looking at the left tail).

### Step 4: Make a Decision
- If the calculated \(z\)-value is less than \(-1.645\), we reject the null hypothesis.
- If the calculated \(z\)-value is greater than or equal to \(-1.645\), we fail to reject the null hypothesis.

### Step 5: Conclusion
Since the specific previous mean completion time (\(\mu_0\)) is not given in the problem, we cannot directly compute the \(z\)-value without this information. However, we can set up the inequality for decision-making:

\[
\frac{82 - \mu_0}{1.875} < -1.645
\]

Solving for \(\mu_0\):

\[
82 - \mu_0 < -1.645 \times 1.875
\]
\[
82 - \mu_0 < -3.084375
\]
\[
\mu_0 > 85.084375
\]

If the historical mean (\(\mu_0\)) is greater than approximately 85.08 days, then there is enough evidence to reject the null hypothesis and support the company's claim that the software has reduced the average project completion time.

### Summary
- If the historical average completion time was more than 85.08 days, there is enough evidence at the 5% significance level to support the company's claim.
- If it was less than or equal to 85.08 days, there is not enough evidence to support the claim.

Do you need further details or clarifications?

Here are 5 related questions to consider:
1. How would the conclusion change if the sample size were smaller, say 36 projects?
2. What is the impact of changing the significance level from 5% to 1% on the test outcome?
3. How would a different population standard deviation affect the result?
4. What would the hypothesis test look like if the population standard deviation were unknown?
5. How does the choice between a one-tailed and two-tailed test influence the conclusion?

**Tip:** Always double-check the assumptions (e.g., normality, known vs. unknown population standard deviation) before choosing the appropriate test statistic for hypothesis testing.

Explore a hypothesis test to determine if a software development company's new project management software reduces project completion time. Includes step-by-step calculations using the z-test formula with known population standard deviation and sample size. Find out if there's enough evidence to support the claim at a 5% significance level.

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