Practice section 7 of 7

$X\sim N(20,\sigma^2)$.

(i) Given $P(12<X<k)=P(X>k)=0.42065$, show that σ=8 to the nearest whole number, and find k. [4]

$X=Y_1+Y_2-4$, where $Y_1,Y_2$ are independent observations of a normal variable Y. W is independent of Y and has normal mean 25 and variance 16.

(ii) Using σ=8, find $E(Y)$ and $\operatorname{Var}(Y)$, then $P(W>2Y)$. [5]

State the parameters of every normal distribution used. Grapefruits and papayas are sold by mass. The table gives mean mass, standard deviation (SD) and price in dollars per gram.

(i) Given 65% of grapefruits weigh more than 300 g, show that μ=306 to the nearest whole number. [2]

Take μ=306 for the remaining parts.

(ii) Find the probability that 3 random grapefruits have total mass below 950 g. [2]

(iii) Ten grapefruits are randomly chosen. Find the probability that more than half weigh over 300 g. [2]

(iv) Find the probability that the total mass of 3 random grapefruits differs from that of 2 random papayas by at most 100 g. [3]

(v) Find the probability that 4 grapefruits cost at least 9 dollars. [3]

(vi) A promotion offers one grapefruit and one papaya for a fixed price. Find the least price a dollars giving at least 90% probability of not making a loss. [3]

Chain-link mass is normal with mean 800 g and standard deviation 20 g.

(i) Find the probability one link exceeds 805 g. [1]

A locking sleeve increases the link’s mass by 10%.

(ii) State the sleeved-link mass distribution and find the probability it lies between 865.35 and 895.5 g. [3]

Hook mass is normal with mean 750 g;15% weigh below 735.6 g.

(iii) Find its standard deviation. [2]

Five independent sleeved links and one hook are packed in a fixed 1 kg wooden box. The probability that the box and contents exceed k grams is more than 0.013.

(iv) Find k to 1 decimal place, stating all distribution parameters used. [4]

Fettuccine packet mass is normal with mean 450 g and standard deviation σg.10% weigh at most 437 g.

(i) Find σ. [2]

Spaghetti packet masses follow $N(500,9.2^2)$. Thomas buys six spaghetti packets and two fettuccine packets.

(ii) Find the probability that the total spaghetti mass differs from twice the two-packet fettuccine total by at least 1250 g. [3]

(iii) State an assumption needed in(ii). [1]

The random variable $R$ is normally distributed such that $R\sim N(\mu,16)$. Find the maximum possible value of $P(-3.74\le R\le5.82)$.