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A recent survey found that 86% of employees plan to devote at least some work time to follow games during the NCAA Men's Basketball Tournament. A random sample of 100 employees was selected. What is the probability that less than 80% of this sample will devote work time to follow games?

User Radhoo
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1 vote

Answer:

4.18% probability that less than 80% of this sample will devote work time to follow games

Explanation:

Normal probability distribution

When the distribution is normal, we use the z-score formula.

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


Z = (X - \mu)/(\sigma)

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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
\mu and standard deviation
\sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
\mu and standard deviation
s = (\sigma)/(√(n)).

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
\mu = p and standard deviation
s = \sqrt{(p(1-p))/(n)}

In this question, we have that:


p = 0.86, n = 100

So


\mu = 0.86, s = \sqrt{(0.86*0.14)/(100)} = 0.0347

What is the probability that less than 80% of this sample will devote work time to follow games?

This is the pvalue of Z when X = 0.8. So


Z = (X - \mu)/(\sigma)

By the Central Limit Theorem


Z = (X - \mu)/(s)


Z = (0.8 - 0.86)/(0.0347)


Z = -1.73


Z = -1.73 has a pvalue of 0.0418

4.18% probability that less than 80% of this sample will devote work time to follow games

User Safwan Samsudeen
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