Answer:
P(X is less than 348) = 0.2148
Step-by-step explanation:
Given that:
Sample proportion (p) = 0.3
Sample size = 1200
Let X be the random variable that obeys a binomial distribution. Then;
[tex]X \sim Bin(n = 1200,p =0.3)[/tex]
The Binomial can be approximated to normal with:
[tex]\mu = np = 1200 \times 0.3 \\ \\ \mu= 360[/tex]
[tex]\sigma = \sqrt{np(1-p) } \\ \\ \sigma = \sqrt{1200 \times (0.3)(1-0.3) } \\ \\ \sigma = 15.875[/tex]
To find:
P(X< 348)
So far we are approximating a discrete Binomial distribution using the continuous normal distribution. 348 lies between 347.5 and 348.5
Normal distribution:
x = 347.5, [tex]\mu[/tex] = 360, [tex]\sigma[/tex] = 15.875
Using the z test statistics;
[tex]z = \dfrac{x - \mu}{\sigma}[/tex]
[tex]z = \dfrac{347.5 - 360}{15.875}[/tex]
[tex]z = \dfrac{-12.50}{15.875}[/tex]
z = -0.7874
z ≅ - 0.79
The p-value for P(X<347.5) = P(Z < -0.79)
From the z tables;
P(X<347.5) = 0.2148
Thus;
P(X is less than 348) = 0.2148
