Respuesta :
Answer:
Without continuity correction, an 89.71% probability of getting less than 25 correct.
With continuity correction, 92.22% probability of getting less than 25 correct.
Step-by-step explanation:
I am going to use the normal approximation to the binomial to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]p = 0.5, n = 40[/tex]
So
[tex]\mu = E(X) = np = 40*0.5 = 20[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{40*0.5*0.5} = 3.1622[/tex]
Less than 25 correct
Without cc
[tex]P(X \leq 24)[/tex], which is the pvalue of Z when X = 24. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{24 - 20}{3.1622}[/tex]
[tex]Z = 1.265[/tex]
[tex]Z = 1.265[/tex] has a pvalue of 0.8971
Without continuity correction, an 89.71% probability of getting less than 25 correct.
With cc
[tex]P(X \leq 24 + 0.5) = P(X \leq 24.5)[/tex], which is the pvalue of Z when X = 24.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{24.5 - 20}{3.1622}[/tex]
[tex]Z = 1.42[/tex]
[tex]Z = 1.42[/tex] has a pvalue of 0.9222
With continuity correction, 92.22% probability of getting less than 25 correct.
