Calculate the 99% confidence interval for the mean length of steel sheets.

The problem is asking for a 99% confidence interval for the population mean of the length of steel sheets. We are given the following information:

Since the population standard deviation is known and the sample size is large (n=64 > 30, or we assume the population is normally distributed as stated in the problem description “Gehen Sie davon aus, dass die Länge der Stahlbleche mit einer normalverteilten Zufallsgröße modelliert ist.”), we can use the z-interval formula for the confidence interval of the population mean:

Where:

For a 99% confidence level, .
Then .
We need to find the z-value such that the area in the upper tail of the standard normal distribution is 0.005. This means the cumulative probability to the left of is .
Looking up the z-table or using a calculator, we find that .

Now we can calculate the margin of error :

Finally, we calculate the confidence interval:

Lower bound:
Upper bound:

We are asked to round the answer to 2 decimal places.
Lower bound rounded to 2 decimal places:
Upper bound rounded to 2 decimal places:

Therefore, the 99% confidence interval is .

Das Konfidenzintervall lautet

Final Answer: The final answer is