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sammy [17]
3 years ago
13

A recent survey found that 86% of employees plan to devote at least some work time to follow games during the NCAA Men's Basketb

all 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?
Mathematics
1 answer:
trasher [3.6K]3 years ago
7 0

Answer:

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

Step-by-step 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 = \frac{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 = \frac{\sigma}{\sqrt{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{\frac{p(1-p)}{n}}

In this question, we have that:

p = 0.86, n = 100

So

\mu = 0.86, s = \sqrt{\frac{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 = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{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

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