Question

3. A national study has shown that 44% of college students engage in binge drinking (5 drinks at one sitting for men, 4 for women). Assume that the 53 students taking Stat 216 this summer represent a random sample of college students in the US.

(a) Define the parameter of interest and what symbol you would use to represent it.

(b) Show whether or not the conditions are met for us to apply the CLT here.

(c) Assuming the conditions are met, what does the Central Limit Theorem say about the sampling distribution of the mean proportion of students that binge drink?

(d) Sketch the sampling distribution of the proportion of 53 college students that engage in binge drinking. (i.e. if we repeatedly sample 53 college students, how would the proportion of binge drinkers be distributed?

(e) What is the probability that less than 30% of students taking Stat 216 this summer engage in binge drinking?

(f) If instead Stat 216 had only 32 students, would it be more or less likely that less than30% of Stat 216 students would engage in binge drinking than for 53 students? Explain

Answer 1

a) Parameter of interest : Population proportion of college students who engage in binge drinking .

Represented by symbol : p

b) Conditions for Central Limit Theorem

i) sample size is large - which is met as n = 53 > 30.

ii) sample size is less than 10% of the population size - which is met as population of college students in US is very large .

iii) sample values must be independent of each other - which is met as each student behave independently.

iv) the data must be sampled randomly - though 53 students from a course is our sample , we assume that this is our random sample .

c) Let be the sample proportion

Then by Central Limit Theorem, sampling distribution of sample proportion follow Normal with

mean = p= 0.44 ( population proportion)

and standard error = =

that is

e) To find

As

then

= P( z < -2.05)

= 0.0202 ( from z table)

Probability that less than 30% of STAT students  engage in binge drinking = 0.0202

f) If n= 32

then

standard error = =

that is

= P( z < - 1.60 )

= 0.0548 ( from z table)

If STAT 216 had only 32 students (sample size is 32), it would be more likely that less than 30% of students would engage in binge drinking than for 53 students .

d)