Practice section 4 of 5

A production manager wants a random quality-control sample of one type of chocolate bar from the thousands produced at a factory in one day. For 70 sampled bars, mass $x$ grams satisfies $\sum(x-95)=287$ and $\sum(x-95)^2=1928$.

(i) State what random sampling means in this context. [1]

(ii) Calculate unbiased population mean and variance estimates. [2]

Fifty random observations are taken from $B(20,0.2)$. Find the probability that the sample mean lies between 3.8 and 4.2. [3]

In the stated probability model, a flu patient has a 98% chance of full recovery. Find the probability that (i) more than 2 in a sample of 20 do not recover fully [2]; (ii) in fifty samples of 20 patients, the mean number not recovering fully is greater than 0.5. [3]

At a fun-fair stall, the probability of winning a prize in each game is 0.05.

(a) Find the largest number of games in a day if the probability of more than 5 prizes is to be less than 0.1. [3]

(b) The stall opens for 70 days, with 60 games each day. Find the probability that the average number of prizes per day is between 2.5 and 3 inclusive. [4]

The last weekday train leaves Bedok at 11.29pm. Its travel time $X$ to Redbridge is normal with mean 30 minutes and variance 9.

(i) Find the travel time exceeded by 90% of trips. [2]

At 11.40pm Abel starts walking to Redbridge; his walking time $Y$ is normal with mean 16 minutes and variance 4.

(ii) Find the probability that the train arrives after midnight and Abel arrives before midnight. [2]

(iii) Find the probability that Abel does not miss the train. [4]

(iv) Explain why(iii) is greater than (ii). [1]

Dina’s walking time $W$ has mean 4 minutes and variance 6.

(v) Explain why $W$ is unlikely to be normal. [1]

(vi) For 40 random trips, find the probability that Dina’s sample mean is below 3.5 minutes. [2]

A Kranji farm sells chickens and ducks.

(a) Chicken mass has mean 2.5 kg and standard deviation 0.4 kg. If the probability that the mean of a large sample of $n$ chickens exceeds 2.45 kg is greater than 0.95, find the least $n$. [3]

(b) Duck mass $X$ kg is normal with mean 2.7 and standard deviation $\sigma$. Ducks cost $4 per kg. (i) Given $P(X<2.5)=\frac{50}{107}P(X<2.9)$, show that $\sigma=0.424$ to 3s.f. [2] (ii) A housewife brings $ 50. Find the probability she has less than $27.50 after buying2 ducks. [3] (iii) For 4 randomly chosen ducks, find the probability the most expensive costs less than $ 11. [2]