Researchers are concerned about the binge drinking behaviors at a local university among members of Greek life organizations. Assume the college reported that binge drinking was observed in 31% of students attending the university. The research team conducts a sample of 213 students involved in a Greek life organization on campus and finds that 94 have reported binge drinking behaviors at least once that year. Can researchers carry out their analysis assuming an approximation to the normal distribution is acceptable?

Calculate the probability of observing the results witnessed in the sample.
Determine the mean and standard deviation for this distribution under the assumption made under the normal approximation. If the Normal approximation is not applicable, state "Not Applicable".

They can carry out their analysis assuming an approximation to the normal distribution because the sample size (n=213) is big enough. It would be very cumbersome to use binomial distribution with that sample size.

To calculate the probability of observing the results witnessed in the sample, we have to define the parameters of the sampling distribution, approximating this to a normal distribution:

[tex]\mu=\pi=0.31\\\\\sigma=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.31(1-0.31)}{213}}=0.032[/tex]

The proportion of the sample is:

[tex]p=94/213=0.44[/tex]

We can calculate the z-value:

[tex]z=\frac{X-\mu}{\sigma} =\frac{0.44-0.31}{0.032}=\frac{0.13}{0.032}=4.062[/tex]

The probability value for z>4.062 is equal to the probability of having a sample mean of 94 or more:

[tex]P(X>94)=P(z>4.062)=0.00002[/tex]