Respuesta :
Answer:
(a) Correct option is (A).
(b) The value of P (X ≥ 770) is 0.0143.
(c) The value of P (X ≤ 720) is 0.0708.
Step-by-step explanation:
Let X = number of elements with a particular characteristic.
The variable p is defined as the population proportion of elements with the particular characteristic.
The value of p is:
p = 0.74.
A sample of size, n = 1000 is selected from a population with this characteristic.
(a)
According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.
The mean of this sampling distribution of sample proportion is:
[tex]\mu_{\hat p}=p[/tex]
The standard deviation of this sampling distribution of sample proportion is:
[tex]\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}[/tex]
The sample selected is of size, n = 1000 > 30.
Thus, according to the central limit theorem the distribution of [tex]\hat p[/tex] is Normal, i.e. [tex]\hat p\sim N(\mu_{\hat p}=0.74,\ \sigma_{\hat p}=0.0139)[/tex].
Thus the correct option is (A).
(b)
We need to compute the value of P (X ≥ 770).
Apply continuity correction:
P (X ≥ 770) = P (X > 770 + 0.50)
= P (X > 770.50)
Then [tex]\hat p> \frac{770.5}{1000}=0.7705[/tex]
Compute the value of [tex]P(\hat p> 0.7705)[/tex] as follows:
[tex]P(\hat p> 0.7705)=P(\frac{\hat p-\mu_{\hat p}}{\sigma_{\hat p}}>\frac{0.7705-0.74}{0.0139})[/tex]
[tex]=P(Z>2.19)\\=1-P(Z<2.19)\\=1-0.98574\\=0.01426\\\approx0.0143[/tex]
Thus, the value of P (X ≥ 770) is 0.0143.
(c)
We need to compute the value of P (X ≤ 720).
Apply continuity correction:
P (X ≤ 720) = P (X < 720 - 0.50)
= P (X < 719.50)
Then [tex]\hat p<\frac{719.5}{1000}=0.7195[/tex]
Compute the value of [tex]P(\hat p<0.7195)[/tex] as follows:
[tex]P(\hat p<0.7195)=P(\frac{\hat p-\mu_{\hat p}}{\sigma_{\hat p}}<\frac{0.7195-0.74}{0.0139})[/tex]
[tex]=P(Z<-1.47)\\=1-P(Z<1.47)\\=1-0.92922\\=0.07078\\\approx0.0708[/tex]
Thus, the value of P (X ≤ 720) is 0.0708.
