Question

You will need to use R to compute the final probabilities, but show all of the work up to that point and write down the R code that you ran to compute your final answer. Problems should be done in the order listed, and should be clear and complete for full marks.

1. The tensile strength of plastic used to make grocery bags is an important quality characteristic. It is known that the strength is normally distributed with mean, μ = 2900 psi and standard deviation, σ = 26 psi. If you purchase 50 of these bags, what is the probability that their mean tensile strength is less than 2890 psi?

2. Starting incomes of IT graduates are roughly normally distributed with μ = $50,000 and σ = $6000. If 12 IT graduates are randomly selected, determine the probability that their mean annual income will exceed $55,000.

Answer 1

If X has normal distribution with mean and standard deviation then the sampling distribution of the sample mean is approximately normally distributed with mean and standard deviation

Where , and

1. Since ,

Therefore ,

Where , and

Now , By using R :

> pnorm(2890,2900,3.6770)
[1] 0.003267964

Therefore , the probability that their mean tensile strength is less than 2890 psi is 0.0032679640.0033

2. Since ,

Therefore ,

Where , and

Now , By using R :

> 1-pnorm(55000,50000,1732.0508)
[1] 0.001946208

Therefore ,  the probability that their mean annual income will exceed $55,000 is 0.0019.