Review practice · Section 1 of 2

A manufacturer launches the environmentally marketed “Green Leaf” car and claims its carbon emission is at most 80 g/km. The transport authority suspects this figure is understated and requests test data from 20 units:

Calculate unbiased estimates of the population mean and variance.2 marks

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 mark

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

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 marks

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 marks; (ii) in fifty samples of 20 patients, the mean number not recovering fully is greater than 0.5.3 marks

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 marks

(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 marks

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 marks

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 marks

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

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

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

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

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