Question Type 1: Finding the mean for a given set of data and distribution with known variance Exercises

In a factory, the weight of widgets is normally distributed with known standard deviation $\sigma=2$ kg. A random sample of 15 widgets has mean weight $\bar{x}=75$ kg. Find the 98% confidence interval for the true mean weight.

A process yields items with a normally distributed diameter. The population standard deviation is known to be $\sigma=5$ mm. A sample of 20 items has mean diameter $\bar{x}=102.5$ mm. Find the 90% confidence interval for the true mean diameter.

A chemical measurement has known population variance $\sigma^2=9$. From a sample of 30 measurements the mean is $\bar{x}=50$. Compute the 99% confidence interval for the true mean.

For a school, exam scores out of 8 are normally distributed with known variance $\sigma^2 = 9$. A sample of 10 students has scores: 4.5, 2.6, 5, 7.6, 2.3, 2.3, 7.7, 5.3, 1.6, 4.6. Find the 95% confidence interval for the population mean $\mu$.

Eight smartphone battery‐life measurements (in hours) are: 10, 11, 9, 12, 10, 11, 9, 10. If the population standard deviation is known to be $\sigma=1.5$, find the 95% confidence interval for the mean battery life.

Ten machine part lengths (in mm) are: 49.5, 50.1, 50.3, 49.8, 50.0, 49.7, 50.2, 49.9, 50.4, 49.6. If the population standard deviation is known to be $\sigma=1.2$ mm, find the 95% CI for the true mean length.

For the same eight battery‐life data (10, 11, 9, 12, 10, 11, 9, 10) but with unknown variance, find the 95% confidence interval for the mean.

A laboratory records 12 readings: the sample mean is $\bar{x}=50$ and the sample standard deviation is $s=2.1$. Assuming normality and unknown population variance, find the 95% confidence interval for the true mean.

A sample of 25 electronic components has mean lifetime $\bar{x}=200$ hours and sample standard deviation $s=10$ hours. Assuming lifetimes are normal with unknown variance, find the 90% confidence interval for the true mean lifetime.

For the same exam scores (4.5, 2.6, 5, 7.6, 2.3, 2.3, 7.7, 5.3, 1.6, 4.6), assume the variance is unknown. Find the 90% confidence interval for the mean $\mu$.

A batch of measurements (n=18) yields $\bar{x}=120$ and $s=4.5$. Assuming normality and unknown variance, compute the 99% confidence interval for the mean.

Using the same ten machine part lengths (49.5, 50.1, 50.3, 49.8, 50.0, 49.7, 50.2, 49.9, 50.4, 49.6) but assuming the variance is unknown, find the 95% CI for the mean.