Answer:
a) The probability that Jodi scores 78% or lower on a 100-question test is 4%.
b) The probability that Jodi scores 78% or lower on a 250-question test is 0.023%.
Step-by-step explanation:
a) To approximate this distribution we have to calculate the mean and the standard distribution.
The mean is the proportion p=0.85.
The standard deviation can be calculates as:
![\sigma=\sqrt{\frac{p(1-p)}{n} }= \sqrt{\frac{0.85*(1-0.85)}{100} }=0.04](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%20%7D%3D%20%5Csqrt%7B%5Cfrac%7B0.85%2A%281-0.85%29%7D%7B100%7D%20%7D%3D0.04)
To calculate the probability that Jodi scores 78% or less on a 100-question test, we first calculate the z-value:
![z=\frac{p-p_0}{\sigma} =\frac{0.78-0.85}{0.04} =-1.75](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bp-p_0%7D%7B%5Csigma%7D%20%3D%5Cfrac%7B0.78-0.85%7D%7B0.04%7D%20%3D-1.75)
The probability for this value of z is
![P(x](https://tex.z-dn.net/?f=P%28x%3C0.78%29%3DP%28z%3C-1.75%29%3D0.04)
The probability that Jodi scores 78% or lower on a 100-question test is 4%.
b) In this case, the number of questions is 250, so the standard deviation needs to be calculated again:
![\sigma=\sqrt{\frac{p(1-p)}{n} }= \sqrt{\frac{0.85*(1-0.85)}{250} }=0.02](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%20%7D%3D%20%5Csqrt%7B%5Cfrac%7B0.85%2A%281-0.85%29%7D%7B250%7D%20%7D%3D0.02)
To calculate the probability that Jodi scores 78% or less on a 250-question test, we first calculate the z-value:
![z=\frac{p-p_0}{\sigma} =\frac{0.78-0.85}{0.02} =-3.5](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bp-p_0%7D%7B%5Csigma%7D%20%3D%5Cfrac%7B0.78-0.85%7D%7B0.02%7D%20%3D-3.5)
The probability for this value of z is
![P(x](https://tex.z-dn.net/?f=P%28x%3C0.78%29%3DP%28z%3C-3.5%29%3D0.00023)
The probability that Jodi scores 78% or lower on a 250-question test is 0.023%.