Question 1185873: The shoulder height for a random sample of six (6) fawns (deer less than 5 months old) in a national park was , 𝑥 = 79.25 cm with population standard deviation 𝞂= 5.33 cm. Compute an 80% confidence interval for the mean shoulder height of the population of all fawns (deer less than 5 months old) in this national park. Analyze the result to interpret its meaning.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **80% Confidence Interval:**
To compute an 80% confidence interval for the mean shoulder height of the population of all fawns in this national park, we can use the following formula:
```
Confidence Interval = x̄ ± z * (σ / √n)
```
Where:
* x̄ is the sample mean (79.25 cm)
* σ is the population standard deviation (5.33 cm)
* n is the sample size (6)
* z is the z-score corresponding to the desired confidence level (80%)
To find the z-score for an 80% confidence level, we need to find the value that leaves 10% in each tail of the standard normal distribution (since 100% - 80% = 20%, and we divide by 2 for the two tails). Using a z-table or calculator, we find that the z-score is approximately 1.28.
Now, we can plug the values into the formula:
```
Confidence Interval = 79.25 ± 1.28 * (5.33 / √6)
```
```
Confidence Interval = 79.25 ± 2.59
```
Therefore, the 80% confidence interval for the mean shoulder height of all fawns in the national park is approximately **(76.66 cm, 81.84 cm)**.
**Interpretation:**
We are 80% confident that the true mean shoulder height of all fawns (deer less than 5 months old) in this national park lies between 76.66 cm and 81.84 cm. This means that if we were to repeatedly take samples of size 6 from the population and calculate 80% confidence intervals, 80% of those intervals would contain the true population mean.
**Analysis:**
The confidence interval provides a range of plausible values for the true population mean. The width of the interval reflects the uncertainty in our estimate. A wider interval indicates more uncertainty, while a narrower interval indicates less uncertainty.
In this case, the 80% confidence interval is relatively narrow, suggesting that we have a fairly precise estimate of the true population mean. However, it is important to note that the sample size is small (n = 6), which can affect the accuracy of the estimate. A larger sample size would generally lead to a narrower confidence interval and a more precise estimate.
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