IB Mathematics Applications & Interpretation Question Type 2: Finding the Confidence Interval for a Given Set of Data and Distribution for Unknown Variance Exercises

A reaction time experiment yields $25$ measurements with sample mean $10.5\text{ s}$ and sample standard deviation $1.2\text{ s}$. Assuming normality with unknown variance, find a $95\%$ confidence interval for the mean reaction time.

A factory measures the diameters of 30 rods, obtaining a sample mean of $10.2\text{ cm}$ and a standard deviation of $0.48\text{ cm}$. Assuming the diameters follow a normal distribution with unknown variance, find a $95\%$ confidence interval for the true mean diameter.

A nutritionist records the weights (kg) of 20 patients. The sample mean is 65 and sample standard deviation is 8. Assuming weights are normally distributed with unknown variance, find the 95% confidence interval for the true mean weight.

For a school, grades out of 8 are assumed normally distributed with unknown variance. A sample of 10 students yields grades 4.5, 2.6, 5.0, 7.6, 2.3, 2.3, 7.7, 5.3, 1.6, 4.6. Find a 90% confidence interval for the mean grade.

Five completion times (in minutes) for a task are 45, 50, 55, 60, and 65. Assuming the times are normally distributed with unknown variance, find a 90% confidence interval for the mean completion time.

Thirteen measurements of a machine’s positional error give a sample mean $0.15\text{ mm}$ and standard deviation $0.05\text{ mm}$. Assuming normality with unknown variance, find a $99\%$ confidence interval for the true mean error.

A sample of 16 measurements yields a mean of 200 and a standard deviation of 25. Find the 90% confidence interval for the true population mean, assuming the population is normally distributed with unknown variance.

A manufacturer tests 12 light bulbs. The lifetimes (in hours) have sample mean $\bar{x}=1200$ and sample standard deviation $s=100$. Assuming normality with unknown variance, find a 95% confidence interval for the true mean lifetime.

Seven observations of a chemical concentration (mg/L) are $100,102,98,101,99,103,97$. Assuming normality with unknown variance, find a $95\%$ confidence interval for the true mean concentration.

Eight test scores are 78, 85, 80, 74, 90, 88, 76, 82. Assuming scores are normal with unknown variance, find a 99% confidence interval for the population mean.

Nine customers’ daily coffee consumption (cups) is $2.1,3.5,2.8,2.9,3.0,2.7,3.2,3.1,2.6$. Assuming normality with unknown variance, find a $95\%$ confidence interval for the mean cups consumed.

Six employees’ annual salaries (in thousands of dollars) are 45, 50, 55, 60, 65, and 70. Assuming salaries are normally distributed with unknown variance, find a 95% confidence interval for the mean salary.