Answer to Question #276173 in Statistics and Probability for kerubo

Question

the weight of a dart to be used in a tournament is normally distributed with a mean of17 grams and a standard deviation of 0.3 grams .find the probability that a dart chosen at random is;

its weight is between 16.55to 17.45 grams

over 18 grams

Answer 1

$\mu=17,\space \sigma =0.3$

Let $X$ be the random variable representing the weight of a dart. Therefore, $X\sim(17,0.3^2)$.

We determine,

$p(16.55\lt X\lt17.45)=p((16.55-17)/0.3\lt Z\lt(17.45-17)/0.3)=p(-1.5\lt Z\lt 1.5)=\phi(1.5)-\phi(-1.5)=0.9332-0.0668=0.8664$

Therefore, the probability that the weight of the selected dart is between 16.55to 17.45 grams is 0.8664.

In the next part, we determine the probability that the weight of the selected dart is over 18 grams given as,

$p(X\gt 18)=p(Z\gt(18-17)/0.3)=p(Z\gt 3.33)=1-p(Z\lt3.33)=1-\phi(3.33)=1-0.9996=0.0004$

Therefore, the probability that the randomly selected dart's weight is over 18 grams is 0.0004.