Question

A student performs an experiment in which a computer is used to simulate drawing a random sample of size p from a larger population. The proportion of the population with the characteristics of interest to the student is p. Let the random variable P represent the sample proportion observed in the experiment. If p=1/5, find the smallest integer value of the sample size such that the standard deviation of P is less than or equal to 0.01 Done, answer n>=1600. My question is from

Given “Each of 23 students in a class independently performs the experiment described and each student calculates an approximate 95% confidence interval for P using sample proportions for their sample. It is subsequently found that exactly one of the 23 confidence intervals calculated by the class does not contain the value of P”

Find “Two of the confidence intervals calculated by the class are selected at random without replacement. Find the probability that exactly one of the selected confidence intervals does not contain the value of P.

Answer 1

95% confidence Interval indicates that out of the total of 23 confidence intervals computed, 95% of them i.e., 23 X 0.95 = 21.85 = 22 (Round to integer) will contain P and 1 will not contain P.

Thus, we have:

Number of confidence Intervals containing P = 22

Number of confidence Intervals not containing P = 1

Total number of confidence intervals = 23

Selection:

Number of confidence Intervals containing P = 1

Number of confidence Intervals not containing P = 1

Total number of confidence intervals = 2

2 confidence intervals can be selected from 23 confidence intervals in 23C2 = 253

1 confidence interval containing P can be selected from 22 confidence intervals containing P in 22C1 = 22

1 confidence interval not containing P can be selected from 1 confidence intervals not containing P in 1C1 = 1

So,

the probability that exactly one of the selected confidence intervals does not contain the value of P = 22 X 1/253 = 0.0870

So,

Answer is:

0.0870