Respuesta :
Answer:
20.90% probability that more than 192 of the confidence intervals cover the true proportions
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:
A single 95% confidence interval has a 95% probability of covering the true proportions, so [tex]p = 0.95[/tex]
200 intervals, so [tex]n = 200[/tex]
Then
[tex]\mu = E(X) = np = 200*0.95 = 190[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{200*0.95*0.05} = 3.0822[/tex]
What is the probability that more than 192 of the confidence intervals cover the true proportions
Using continuity correction, this is [tex]P(X > 192 + 0.5) = P(X > 192.5)[/tex], which is 1 subtracted by the pvalue of Z when X = 192.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{192.5 - 190}{3.0822}[/tex]
[tex]Z = 0.81[/tex]
[tex]Z = 0.81[/tex] has a pvalue of 0.7910
1 - 0.7910 = 0.2090
20.90% probability that more than 192 of the confidence intervals cover the true proportions
