Respuesta :
Answer:
(a) Probability that fewer than 100 will be under 18 years of age is 0.19655.
(b) Probability that 200 or more will be over 59 years of age is 0.00449.
Step-by-step explanation:
We are given that over 70% of households play such games. Of those individuals who play video and computer games, 18% are under 18 years old, 53% are 18-59 years old, and 29% are over 59 years old.
(a) A sample of 600 people is selected and we have to find the probability that fewer than 100 will be under 18 years of age.
The z score probability distribution for sample proportion is given by;
Z = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion of people = [tex]\frac{100}{600}[/tex] = 16.67%
p = population proportion of people under 18 years old = 18%
n = sample of people = 600
Now, probability that fewer than 100 will be under 18 years of age is given by = P( [tex]\hat p[/tex] < 0.167)
P( [tex]\hat p[/tex] < 0.167) = P( [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < [tex]\frac{0.167-0.18}{\sqrt{\frac{0.167(1-0.167)}{600} } }[/tex] ) = P(Z < -0.854) = 1 - P(Z [tex]\leq[/tex] 0.854)
= 1 - 0.80345 = 0.19655
The above probability is calculated by looking at the value of x = 0.854 in the z table which will lie between x = 0.85 and x = 0.86.
(b) A sample of 800 people is selected and we have to find the probability that 200 or more will be over 59 years of age.
The z score probability distribution for sample proportion is given by;
Z = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion of people = [tex]\frac{200}{800}[/tex] = 25%
p = population proportion of people over 59 years old = 29%
n = sample of people = 800
Now, probability that 200 or more will be over 59 years of age is given by = P( [tex]\hat p[/tex] [tex]\geq[/tex] 0.25)
P( [tex]\hat p[/tex] [tex]\geq[/tex] 0.25) = P( [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] [tex]\geq[/tex] [tex]\frac{0.25-0.29}{\sqrt{\frac{0.25(1-0.25)}{800} } }[/tex] ) = P(Z [tex]\geq[/tex] -2.613) = P(Z [tex]\leq[/tex] 2.613)
= 1 - 0.99551 = 0.00449
The above probability is calculated by looking at the value of x = 2.613 in the z table which will lie between x = 2.61 and x = 2.62.
