During a recent drought, a water utility in a certain town sampled 100 residential water bills and found that 73 of the residences had reduced their water consumption over that of the previous year. f) If 95% confidence intervals are computed for 200 towns, what is the probability that more than 192 of the confidence intervals cover the true proportions

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:

The standard deviation of the binomial distribution is:

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

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 , .

In this problem, we have that:

A single 95% confidence interval has a 95% probability of covering the true proportions, so

200 intervals, so

Then

What is the probability that more than 192 of the confidence intervals cover the true proportions

Using continuity correction, this is , which is 1 subtracted by the pvalue of Z when X = 192.5. So

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

30005

yes ok