Practice section 2 of 5

If a random sample of size 60 is taken from $X\sim B(9,0.5)$, find the probability that the sample mean exceeds 5.

A vineyard produces an average of 400 barrels of wine per day, with standard deviation 70.

(a) For a random sample of 50 days’ production, is $\bar X$ normally distributed? Explain briefly and state its mean and variance.

(b) Find the probability that the sample mean is more than 380.

Two firms A and B manufacture components with mean breaking strengths 6 kN and 5.5 kN and standard deviations 0.4 kN and 0.2 kN respectively. Random samples of 100 components from A and 50 from B are tested. Find the probability that the mean strength from A is between 0.45 kN and 0.55 kN more than that from B.

The distance between a pupil’s home and school is $x$ km. A random sample of 60 pupils gives $\sum x=33.6$ and $\sum x^2=22.79$. Calculate unbiased estimates of the population mean and variance.

Estimate the probability that the mean value of $x$, for this sample, will be less than 0.5. State any assumptions or approximations used.

The diameters, in mm, of 80 rods selected at random from a large consignment are summarised below.

Find unbiased estimates of the population mean and variance of the rod diameters.

Find unbiased estimates of the population mean and variance from each sample.

(a) $\sum x=15508$, $\sum x^2=2413891$, $n=100$.

(b) $\sum(t-36.9)=-12$, $\sum(t-36.9)^2=79.21$, $n=40$.