A team of geologists plans to measure the weights of 300 roc

A team of geologists plans to measure the weights of 300 rocks. After weighing each rock a large number of times, they will compute a 99% confidence interval for its weight. Assume there is no bias in the weighing procedure. What is the probability that more than 285 of the confidence intervals will cover the true weights of the rocks? The probability that more than 285 of the confidence intervals will cover the true weights of the rocks is

Solution

let X be the number of confidence intervals which will cpver the true weights of the rocks.

there are 300 rocks. hence there are 300 confidence intervals.

99% confidence interval means the probability that the confidence interval will contain true weight is 0.99

hence X~Bin(300,0.99)

but as 300 is quite a large number. hence normal approximation can be applied.

now E[X]=300*0.99=297 V[X]=300*0.99*(1-0.99)=2.97

hence X can be approximated to follow a normal distribution with mean 297 and variance 2.97

now P[X>285]=1-P[X<285]=1-0.00001=0.99973 [answer]