Researchers published a study in which they considered the incidence among the elderly of various mental health conditions such as dementia, bi-polar disorder, obsessive compulsive disorder, delirium, and Alzheimer's disease. In the U. S., 45% of adults over 65 suffer from one or more of the conditions considered in the study. Calculate the probability that fewer than 320 out of the n

The step-by-step procedure to solve this question is given, using the normal approximation to the binomial distribution.

Step-by-step explanation:

We use the normal approximation to the binomial distribution 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 distributions 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 the U.S., 45% of adults over 65 suffer from one or more of the conditions considered in the study.

This means that

Mean and standard deviation for the approximation, for a sample of n:

Calculate the probability that fewer than 320 out of the n

Using continuity correction, this is , which is the pvalue of Z when X = 319.5. So

Just have to replace the value of n, find Z, and then find its pvalue.

c

step-by-step explanation:

because that’s us why