Question

We believe that 82% of the population of all business statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the population. If the true percentage is really 82%, find the probability of observing 38 or more students who consider statistics to be an exciting subject

Answer 1

Answer:

The probability of observing 38 or more students liking Business statistics is 0.0041 or 0.41%.

Step-by-step explanation:

In the question given, the number of students undergone trial = 39

No. of trials made  

No. of students undergone trial = 37 = n

Let the event of a student tested being interested in the subject of business statistics be a success  

Probability of success in each trial  

= probability that a student tested is interested in the subject of statistics  

= p = 82/100 = 0.82

Therefore,  

q = 1 - p  

q = 18/100 = 0.18

Let "s" indicate the number of successes in "n" trials  

since the number of students who are interested in statistics are in the range of 0 to 39

The values "s" can be any of 0, 1, 2, .... 39  

"S" is a discrete random variate with range = {0, 1, 2, .... 39}  

The probability distribution of "s" is a binomial distribution..  

(Because there a number of trials, all of which are same and independent with only two outcomes in each trial either a success or failure.

Success is the student being interested or a failure, the student not being interested.

The probability of success in each trial remaining the same all through out)

The probability function of a binomial distribution  

==> P(S=s) = n$C_{s}$ * p^s * q^(n-s)  

Therefore, for the above distribution  

P(S=s) = 39$C_{s}$ * (0.82)^s * (0.18)^(39-s)  

Therefore,  

Probability of observing 38 or more of the students who consider business statistics to be an exciting subject  

==> P(S >= 38) = P(S=38) + P(S=39)  

P(S = 38) = 39$C_{38}$ * (0.82)^38 * (0.18)  

= 39 * (0.82)^38 * (0.18) = 0.0037

P(S = 39) = 39$C_{39}$ * (0.82)^39 * (0.18)^0 = (0.82)^39 = 0.0004

Hence the probability of observing 38 or more students liking Business statistics is 0.0041 or 0.41%.