Average human gestation time is 266 days long, when counted from conception. A hospital gives a 90% confidence interval for the mean gestation time from conception among its patients. that interval is 264 +/- 5 days. Is the mean gestation time in that hospital significantly different from 266 days?

To determine if the mean gestation time in the hospital is significantly different from 266 days, we can conduct a hypothesis test using the confidence interval provided. Here’s a step-by-step approach to analyzing this:

### Step 1: Define the Hypotheses

- **Null Hypothesis (\(H_0\))**: The mean gestation time is 266 days.
- **Alternative Hypothesis (\(H_a\))**: The mean gestation time is not 266 days.

### Step 2: Interpret the Confidence Interval

The 90% confidence interval given is \(264 \pm 5\) days, which translates to the interval \([259, 269]\) days. This interval means that we are 90% confident that the true mean gestation time for patients in this hospital lies between 259 and 269 days.

### Step 3: Determine Significance

The confidence interval \([259, 269]\) does include the value 266. This means there is no evidence to reject the null hypothesis at the 90% confidence level, as the true mean gestation time could be 266 days. Therefore, the mean gestation time in this hospital is not significantly different from 266 days based on the provided confidence interval.

### Step 4: Conclusion

Since 266 days is within the 90% confidence interval, we do not have sufficient evidence to conclude that the mean gestation time is significantly different from 266 days at the 90% confidence level.

### Further Analysis
If you wanted to test this hypothesis at a higher confidence level (e.g., 95%), you would need to calculate a new confidence interval. If that interval also contains 266, the result remains the same.

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### Would you like more details or have any questions?

Here are some related questions to explore further:

1. How is a confidence interval calculated?
2. What factors can affect the width of a confidence interval?
3. How does the sample size influence the confidence interval?
4. What is the relationship between confidence level and statistical significance?
5. How can hypothesis testing be used to make decisions in other fields?
6. What are Type I and Type II errors in hypothesis testing?
7. How does the choice of confidence level affect hypothesis testing conclusions?
8. Can we conclude anything about the variance of gestation times from this information?

**Tip:** Remember, the significance of a hypothesis test depends on whether the value being tested falls within the confidence interval, not just on whether it is close to the mean.

Explore whether the mean gestation time in a hospital is significantly different from 266 days using hypothesis testing and confidence intervals. Understand how to interpret confidence intervals and determine statistical significance.

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