Respuesta :
Answer:
The required probability is calculated as 0.052
Solution:
As per the question:
The probability that people dropped out in the first 4 weeks of the program, p = 0.25
The size of the sample, n = 246
Now,
To calculate the Probability of at least 195 people being in the program after first 4 weeks:
[tex]P(X\geq 195) = P(\frac{X - \mu}{\sigma})[/tex] (1)
where
[tex]\mu[/tex] = mean
[tex]\sigma[/tex] = standard deviation
X = No. of people still part of the program
Now,
Mean can be given as:
[tex]\mu = np = 0.25\times 246 = 61.5 = 62\ (approx)[/tex]
The mean no. of people still part of the program = 246 - 62 = 184
Standard deviation is given by:
[tex]\sigma = \sqrt{npq} = \sqrt{np(1 - p)}[/tex]
where
q = 1 - p = 1 - 0.25 = 0.75
[tex]\sigma = \sqrt{246\times 0.25\times 0.75} = 6.79[/tex]
Now, using the appropriate values in eqn (1):
[tex]P(X\geq 195) = P(\frac{X - \mu}{\sigma}\geq \frac{195 - 184}{6.79})[/tex]
[tex]P(X\geq 195) = P(Z\geq 1.62)[/tex]
[tex]P(X\geq 195)[/tex] = 1 - P(Z < 1.62)
Using Z-table:
[tex]P(X\geq 195) = 1 - 0.94738 = 0.052[/tex]
Thus the required probability is calculated as 0.052
