Topic 4.16: Confidence intervals (HL only) Questions

Keeping everything else fixed, increasing the confidence level from 90% to 99% makes the interval:

Which formula gives a confidence interval for a population mean when the population variance is unknown?

A GDC t-interval for a sample of 18 measurements should be run with how many degrees of freedom?

A poll estimates the mean number of hours per week teenagers spend exercising. Using x̄ = 5.0 hours and sₙ₋₁ = 1.5 hours, one researcher uses a sample of n = 25 and another uses n = 100. For a 95% interval, explain which researcher gets the narrower interval, and roughly by what factor.

A transport company records the morning commute time (minutes) for a random sample of 30 employees: x̄ = 42.0 min with unbiased standard deviation sₙ₋₁ = 6.5 min. Find a 95% confidence interval for the mean commute time of all employees.

A 95% confidence interval for a machine's mean output is (47.2, 50.8) units. A manager claims the true mean is 49 units. The data:

Two 95% confidence intervals for the same mean are built from samples of size 20 and size 80 (same sample SD). Which is narrower, and why?

The masses (grams) of a random sample of 10 eggs are: 58, 61, 59, 63, 60, 57, 62, 64, 58, 60. (a) Find the unbiased estimate of the mean and of the standard deviation. (b) Find a 95% confidence interval for the mean egg mass.

A protein bar is advertised as containing 21 g of protein. A consumer test of 40 bars gives x̄ = 20.4 g with sₙ₋₁ = 1.2 g. (a) Find a 95% confidence interval for the mean protein content. (b) Comment on whether the data support the 21 g claim.

An ecologist measures the dissolved-oxygen level (mg/L) of a river at 15 randomly chosen points: x̄ = 8.20 mg/L, sₙ₋₁ = 0.45 mg/L. (a) Find a 95% confidence interval for the mean dissolved-oxygen level. (b) State one assumption needed for this interval to be valid.

A quality engineer samples 12 light bulbs and records lifetimes with x̄ = 1180 hours and sₙ₋₁ = 85 hours. Find the 90% and the 99% confidence intervals for the mean lifetime, and state which is wider.