Answer to Question #317720 in Statistics and Probability for Student2014_

Question

We’ve all had those annoying cell phone calls from Heather, Daisy, or Oscar, trying to sell life insurance or an additional car warranty. First Orion, a call protection agency, recently issued a report that suggested 45% of all cell phone calls in 2019 will be spam. Suppose 500 cell phone calls are selected at random. Use the normal approximation to calculate the probability that less than 100 of cell phone calls will be sparm.

Answer to 4 decimal points.

Answer 1

$p=0.45\\Z=\sqrt{n}\frac{\hat{p}-p}{\sqrt{p\left( 1-p \right)}}\sim N\left( 0,1 \right) \\P\left( \hat{p}\leqslant \frac{100}{500} \right) =P\left( \hat{p}\leqslant 0.2 \right) =P\left( \sqrt{500}\frac{\hat{p}-0.45}{\sqrt{0.45\left( 1-0.45 \right)}}\leqslant \sqrt{500}\frac{0.2-0.45}{\sqrt{0.45\left( 1-0.45 \right)}} \right) =\\=P\left( Z\leqslant -11.2367 \right) =\varPhi \left( -11.2367 \right) =1.35\cdot 10^{-29}$