Probability & Statistics 2 — Variant 3

Anton believes that $10\%$ of students at his college are left-handed. Aliya believes that this is an underestimate. She plans to carry out a hypothesis test of the null hypothesis $p = 0.1$ against the alternative hypothesis $p > 0.1$, where $p$ is the actual proportion of students at the college that are left-handed. She chooses a random sample of $20$ students from the college. She will reject the null hypothesis if at least $5$ of these students are left-handed.

Explain what is meant by a Type I error in this context.

Find the probability of a Type I error in the test.

Given that the true value of $p$ is $0.3$, find the probability of a Type II error in the test.

Each box of Seeds \& Raisins contains $S$ grams of seeds and $R$ grams of raisins. The weight of a box, when empty, is $B$ grams. $S$, $R$ and $B$ are independent random variables, where $S \sim \mathrm{N}(300, 45)$, $R \sim \mathrm{N}(200, 25)$ and $B \sim \mathrm{N}(15, 4)$. A full box of Seeds \& Raisins is chosen at random.

Find the probability that the total weight of the box and its contents is more than $500$ grams.

Find the probability that the weight of seeds in the box is less than $1.4$ times the weight of raisins in the box.