Suppose 30% of a population possess a given characteristic. If a random sample of size 1200 is drawn from the population, then the probability that less than 348 possess that characteristic is

The population follows a normal distribution.

The probability that less than 348 possess that characteristic is 0.2248

The given parameters are:

Start by calculating the mean:

Calculate the standard deviation

Calculate the z-score

Where:

x = 348

So, we have:

So, the probability is represented as:

From the z-table of probabilities, we have:

Hence, the probability that less than 348 possess that characteristic is 0.2248

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The probability that less than 348 possess that characteristic is 0.2148 = 21.48%.

Step-by-step explanation:

I am going to use the binomial approximation to the normal 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 question, we have that:

So

The probability that less than 348 possess that characteristic is

Using continuity correction, this is P(X < 348 - 0.5) = P(X < 347.5), which is the pvalue of Z when X = 347.5. So

has a pvalue of 0.2148.

The probability that less than 348 possess that characteristic is 0.2148 = 21.48%.

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