A company uses a machine to pack bird seed into packets. Packets should contain a mean mass of \(400

Question

A company uses a machine to pack bird seed into packets. Packets should contain a mean mass of $400$ grams of seed. Following maintenance to the packing machine, the owner of the company wishes to check if packets contain a mean mass of $400$ grams of seed. He decides to carry out a hypothesis test at the $5\%$ level of significance. The owner finds the masses, $x$ grams, of bird seed in a random sample of packets. Summary data is as follows. $n=40\qquad\qquad\sum(x-400)=120\qquad\qquad\sum(x-400)^2=1902$ Write down the hypotheses for the test, defining the parameter(s) you use.[2] Find the critical region for the test, giving your answer in grams correct to 1 decimal place.[4] State the conclusion of the test in the context of the question.[2] The company also has a machine that fills bags with pet food. The mean mass of pet food is stated to be $500$ grams per bag. Following comments from customers, the owner of the company decides to carry out a $z$-test using $\mathrm{H}_0:\mu=500$ $\mathrm{H}_1:\mu>500$. State what the owner is testing for.[2] State the critical $z$-value used for this test when it is carried out at the $5\%$ level of significance.[1] The machine is used each week from Monday to Friday. The owner decides to take the last $5$ bags filled on Monday, Wednesday and Friday one week, to make up a sample of $15$ bags for his test. Give two reasons why the owner's sample is not suitable for this test.[2]

Accepted Answer

(a)(i) $\mathrm{H}_0:\mu=400$, $\mathrm{H}_1:\mu\neq 400$, where $\mu$ grams is the population mean mass of seed in a packet (ii) $\overline{x}\leq 398.1$ or $\overline{x}\geq 401.9$ (iii) $\overline{x}=403$ lies in the critical region, so $\mathrm{H}_0$ is rejected: at the $5\%$ level there is sufficient evidence that the mean mass of seed in a packet is not $400$ grams (b)(i) Whether the mean mass of pet food per bag is greater than the stated $500$ grams (ii) $1.645$ (iii) The sample is not random, as only the last $5$ bags on three days are taken. The sample of $15$ is too small for the Central Limit Theorem, and the masses are not known to be normally distributed

N2026 P2 Q11

N2026 P2 Q11