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nevsk [136]
3 years ago
8

An IAB study on the state of original digital video showed that original digital video is becoming increasingly popular. Origina

l digital video is defined as professionally produced video intended only for ad-supported online distribution and viewing. According to IAB data, 30% of American adults 18 or older watch original digital videos each month.
1. Suppose that you take a sample of 100 U.S. adults, what is the probability that fewer than 25 in your sample will watch original digital videos?
Mathematics
1 answer:
never [62]3 years ago
5 0

Answer:

11.51% probability that fewer than 25 in your sample will watch original digital videos

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:

E(X) = np

The standard deviation of the binomial distribution is:

\sqrt{V(X)} = \sqrt{np(1-p)}

Normal probability distribution

Problems of normally distributed samples can be solved using 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.

When we are approximating a binomial distribution to a normal one, we have that \mu = E(X), \sigma = \sqrt{V(X)}.

In this problem, we have that:

n = 100, p = 0.30

So

\mu = E(X) = np = 100*0.3 = 30

\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.3*0.7} = 4.58

What is the probability that fewer than 25 in your sample will watch original digital videos?

Using continuity correction, this is P(X < 25 - 0.5) = P(X < 24.5), which is the pvalue of Z when X = 24.5. So

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

Z = \frac{24.5 - 30}{4.58}

Z = -1.2

Z = -1.2 has a pvalue of 0.1151

11.51% probability that fewer than 25 in your sample will watch original digital videos

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