Inferential Statistics · Paper 1 — Free Maths AA HL Practice Questions

A factory produces metal rods with lengths that are normally distributed with a standard deviation of $0.2\,\text{cm}$. The production manager claims that the mean length of the rods is $15.0\,\text{cm}$. A quality control inspector suspects that the mean length is less than $15.0\,\text{cm}$. He measures a random sample of $10$ rods and finds a sample mean of $14.9\,\text{cm}$.

A pharmaceutical company claims that a new drug reduces the mean recovery time for a certain illness to less than 7 days. A medical researcher tests this claim at the $5\%$ significance level. The recovery times, in days, for a random sample of 10 patients are: $$6.2,\ 7.1,\ 5.8,\ 6.5,\ 6.9,\ 7.3,\ 5.5,\ 6.8,\ 6.1,\ 6.7$$ Recovery times are assumed to be normally distributed with unknown variance.

A pharmaceutical company has developed a new drug intended to reduce systolic blood pressure. In a clinical trial, a random sample of $n = 10$ patients had their systolic blood pressure measured before and after treatment. The reductions in systolic blood pressure (in mmHg) are: $$12.5,\quad 8.2,\quad 15.1,\quad 9.8,\quad 11.4,\quad 13.6,\quad 7.9,\quad 14.2,\quad 10.5,\quad 16.3$$ Assume the reductions are normally distributed with unknown mean $\mu$ and unknown variance $\sigma^2$.