Download n = 10 , p = 1/2

Multiple choice test – 10 True/False questions:
n = 10, p = 1/2
Using the probability rule we can say that:
In a test with these characteristics it is usual to guess correctly anywhere from 2 to 8 questions.
Find the mean and the standard deviation of the binomial distribution
The mean is mu = np = 10 (1/2) = 5
The standard deviation is sigma = sqrt(n*p*q) = sqrt(10(1/2)(1/2)) = 1.58 ~ 1.6
Range rule of thumb:
[mu – 2*sigma; mu + 2*sigma] = [5-2(1.6) ; 5+2(1.6)] = [1.8 ; 8.2]
According to the range rule of thumb it is usual to guess from 2 to 8 questions correctly
Probability of passing the test = .20508 + .11719 + .04395 + .00977 + .00098 = .37697
Is the normal approximation appropriate? Are both, n*p and n*q greater than or equal to 5?
YES!!! P(passing) = normalcdf(5.5, 10^9, 5, 1.264911) = 0.3759
Multiple choice test – 10 three-choices questions:
n = 10, p = 1/3
Using the probability rule we can say that:
In a test with these characteristics it is usual to guess correctly anywhere from 1 to 6 questions.
Find the mean and the standard deviation of the binomial distribution
The mean is mu = np = 10 (1/3) = 3.3
The standard deviation is sigma = sqrt(n*p*q) = sqrt(10(1/3)(2/3)) = 1.45 ~ 1.5
Range rule of thumb:
[mu – 2*sigma; mu + 2*sigma] = [3.3-2(1.3) ; 3.3+2(1.3)] = [0.3 ; 6.3]
According to the range rule of thumb it is usual to guess from 1 to 6 questions correctly
Probability of passing the test = 0.0569 + .01626 + .00305 + .00034 + .000017 = .0766
Is the normal approximation appropriate? Are both, n*p and n*q greater than or equal to 5?
NO!
n*p = 10*(1/3) = 3.33 < 5
Multiple choice test – 10 five-choices questions:
n = 10, p = 1/5
Using the probability rule we can say that:
In a test with these characteristics it is usual to guess correctly anywhere from 0 to 4 questions.
Find the mean and the standard deviation of the binomial distribution:
The mean is mu = np = 10 (1/5) = 2
The standard deviation is sigma = sqrt(n*p*q) = sqrt(10(1/5)(4/5)) = 1.26 ~ 1.3
Range rule of thumb:
[mu – 2*sigma; mu + 2*sigma] = [2-2(1.3) ; 2+2(1.3)] = [-.6 ; 4.6]
According to rule of thumb it is usual to guess from 0 to 4 questions correctly
Probability of passing the test = .00551 + .00079 + .000074 + .0000041 + .0000001 = .0064
Is the normal approximation appropriate? Are both, n*p and n*q greater than or equal to 5?
NO!
n*p = 10*(1/3) = 3.33 < 5