Respuesta :
Answer:
(a) The distribution of the sample proportion is Normal distribution.
(b) The probability that in this sample of 150 households that more than 50% own a 4K television is 0.00012.
Step-by-step explanation:
We are given that in the United States, 35% of households own a 4K television.
Suppose we take a random sample of 150 households.
Let [tex]\hat p[/tex] = sample proportion of households who own a 4K television.
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
p = population proportion of households own a 4K television = 35%
n = sample of households = 150
(a) The distribution of the sample proportion is related to the Normal distribution.
(b) Probability that in this sample of 150 households more than 50% own a 4K television is given by = P( [tex]\hat p[/tex] > 0.50)
P( [tex]\hat p[/tex] > 0.50) = P( [tex]\frac{ \hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] > [tex]\frac{ 0.50-0.35}{\sqrt{\frac{0.50(1-0.50)}{150} } }[/tex] ) = P(Z > 3.67) = 1 - P(Z [tex]\leq[/tex] 3.67)
= 1 - 0.99988 = 0.00012
Now, in the z table the P(Z [tex]\leq[/tex] x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 3.67 in the z table which has an area of 0.99988.
Therefore, probability that in this sample of 150 households more than 50% own a 4K television is 0.00012.
