Question

The scores of individual students on the American College Testing (ACT) Program College Entrance Exam have a normal distribution with mean 18.6 and standard deviation 6.0. At Westside High, 400 students take the test. Assume their scores are all independent. If the scores at this school have the same distribution as nationally, the probability that the average of the scores of all 400 students exceeds 19.0 is smaller than the probability that a single student has a score exceeding 19.0 larger than the probability that a single student has a score exceeding 19.0 the same as the probability that a single student has a score exceeding 19.0

Answer 1

The probability that the average of the scores of all 400 students exceeds 19.0 is larger than the probability that a single student has a score exceeding 19.0

Explanation:

Xi~N(18.6, 6.0), n=400, Yi~Ber(p); Z~N(0, 1);

`P(0\leq X\leq 19.0)=P((0-\mu)/(\sigma) \leq (X-\mu)/(\sigma)\leq (19-\mu)/(\sigma)), Z= (X-\mu)/(\sigma), \mu=18.6, \sigma=6.0`

`P(-3.1\leq Z\leq 0.0667)=\Phi(0.0667)-\Phi (-3.1)=\Phi(0.0667)-(1-\Phi (3.1))=0.52790+0.99903-1=0.52693`

P(Xi≥19.0)=0.473

`\{Yi=0, Xi<  19\\Yi=1, Xi\geq  19\}`

p=0.473

Yi~Ber(0.473)

`P((1)/(n)\displaystyle\sum_(i=1)^(n)X_i\geq 19)=P(\displaystyle\sum_(i=1)^(400)X_i\geq 7600)`

Based on the Central Limit Theorem:

`\displaystyle\sum_(i=1)^(n)X_i\~{}N(n\mu, √(n)\sigma),\displaystyle\sum_(i=1)^(400)X_i\~{}N(7440, 372)`

Then:

`P(\displaystyle\sum_(i=1)^(400)X_i\geq 7600)=1-P(0<\displaystyle\sum_(i=1)^(400)X_i<7600)=\\1-P(-20<Z<0.43)=1-(\Phi(0.43)-(1-\Phi(20)))=0.334`

`P(\displaystyle\sum_(i=1)^(n)Y_i=1)=P(\displaystyle\sum_(i=1)^(400)Y_i=1)`

Based on the Central Limit Theorem:

`\displaystyle\sum_(i=1)^(400)Y_i\~{}N(400* 0.473, √(400)* 0.499)=\displaystyle\sum_(i=1)^(400)Y_i\~{}N(189.2; 9.98)`

`P(\displaystyle\sum_(i=1)^(400)Y_i=1)\~{=}P(0.5<\displaystyle\sum_(i=1)^(400)Y_i<1.5)=P(-18.9<Z<-18.8)\~{=}0`

Then:

the probability that the average of the scores of all 400 students exceeds 19.0 is larger than the probability that a single student has a score exceeding 19.0