Answer to Question #197562 in Statistics and Probability for jack

Question

The latest M1 Macbook Air and Macbook Pro are available for sale at the Apple Store in Gulshan. 70% of the people who want to buy a new Macbook prefer the Macbook Air, while the rest prefer the Macbook Pro. Suppose 16 customers were randomly selected among those shopping for Macbooks on a specific day.What is the probability that the number of people who want a Macbook Air is more than two standard deviations away from the mean value?

Answer 1

Let p be the proportion of people who buy MacBook Air

 Then given p=0.70

Let X denote the number of people Who buy MacBook Air. If 16 customers are selected then-

   $X\sim Bin(16,0.7)$

Required Probability:

$P(x-\mu_x>2\sigma_x)=P(x-11.2>2\times 1.833)$

          $=P(X>14.866) \\ =P(x=15)+P(X=16)\\=^{16}C_{15}(0.7)^{15}(0.3)+^{16}C_{16} (0.7)^{16} \\ =0.0228+.0033\\ =0.0261$

             

Hence The probability that the number of people who want a MacBook Air is more than two standard deviations away from the mean value is 0.0261