Dean Chalmers and Julian Gilbey Solutions for Chapter: The Normal Distribution, Exercise 8: EXERCISE 8D

The masses, $w$ grams, of a large sample of apples are normally distributed with mean $200$ and variance $169.$ Given that the masses of $3413$ apples are in the range $187⩽w<213,$ calculate an estimate of the number of apples in the sample.

The ages of the children in a gymnastics club are normally distributed with mean $15.2$ years and standard deviation $\sigma .$ Find the value of $\sigma$ given that $30.5\%$ of the children are less than $13.5$ years of age.[Use ${\Phi }^{-1}(0.695)=0.51$]

The speeds, in ${\mathrm{kmh}}^{-1},$ of vehicles passing a particular point on a rural road are normally distributed with mean $\mu$ and standard deviation $20.$ Find the value of $\mu$ and find what percentage of the vehicles are being driven at under $80{\mathrm{kmh}}^{-1},$ given that $33\%$ of the vehicles are being driven at over $100 {\mathrm{kmh}}^{-1}.$[Use $\Phi (0.56)=0.722$]

Coffee beans are packed into bags by the workers on a farm, and each bag claims to contain $200 \mathrm{g}.$ The actual mass of coffee beans in a bag is normally distributed with mean $210 \mathrm{g}$ and standard deviation $\sigma .$ The farm owner informs the workers that they must repack any bag containing less than $200 \mathrm{g}$ of coffee beans. Find the value of $\sigma ,$ given that $0.5\%$ of the bags must be repacked.[Use ${\Phi }^{-1}(0.995)=2.576$]

Colleen exercises at home every day. The length of time she does this is normally distributed with mean $12.8$ minutes and standard deviation $\sigma .$ She exercises for more than $15$ minutes on $42$ days in a year of $365$ days.

Calculate the value of $\sigma .$[Use ${\Phi }^{-1}(0.885)=1.2$]

Colleen exercises at home every day. The length of time she does this is normally distributed with mean $12.8$ minutes and standard deviation $\sigma .$ She exercises for more than $15$ minutes on $42$ days in a year of $365$ days.

On how many days in a year would you expect Colleen to exercise for less than $10$ minutes?

The times taken by $15$ -year-olds to solve a certain puzzle are normally distributed with mean $\mu$ and standard deviation $7.42$ minutes.

Find the value of $\mu ,$ given that three-quarters of all $15$ -year-olds take over $20$ minutes to solve the puzzle.[Use ${\Phi }^{-1}(0.75)=0.674$]

The times taken by $15$ -year-olds to solve a certain puzzle are normally distributed with mean $\mu$ and standard deviation $7.42$ minutes.

Find the value of $\mu ,$ given that three-quarters of all $15$ -year-olds take over $20$ minutes to solve the puzzle.