suppose it iss known that 40 of radio listeneres at a partic

suppose it iss known that 40% of radio listeneres at a particuar college are smokers. A sample of 400 students from the college is selected at random. Approximate the probability that at least 180 of these students are radio listeners.

Solution

We are given that 40% of radio listeneres at a particuar college are smokers.

that is it is known that p = 40% = 40/100 = 0.4

number of students from college (n) = 400

We have to calculate probability that at least 180 of these students are radio listeners.

sample proportion (p^) = number of students are radio listeners / n

p^ = 180 / 400 = 0.45

We have to find P(p^ > 0.45)

We can find this probability by using normal distribution.

We know that sample proportion (p^) follows normal distribution with mean p and standard deviation (SD) is

sqrt( (p*q)/n ).

where q = 1 - p

q = 1 - 0.4 = 0.6

SD = sqrt ((p*q)/n ) = sqrt ((0.4*0.6)/400) = 0.0245

P( (p^-p) / sqrt((p*q)/n) > (0.45 - p) / sqrt((p*q)/n) ) = P(Z > (0.45 - 0.4) / sqrt((0.4*0.6) /400)

P(Z > 2.0412) = 1 - P(Z <=2.0412)

This probability we can find by using EXCEL.

Statistical table and EXCEL always gives left tail probability.

syntax :

=NORMSDIST(z)

where, z is test statistic value = 2.0412

P(Z <=2.0412) = 0.9794

P(Z > 2.0412) = 1 - 0.9794 = 0.0206