Consider the probability that more than 97 out of 120 people will get the flu this winter. Assume the probability that a given person will get the flu this winter is 97%. Specify whether the normal curve can be used as an approximation to the binomial probability by verifying the necessary conditions.

Since , the normal distribution cannot be used as an approximation to the binomial probability to approximate the probability.

Using the binomial distribution, 100% probability that more than 97 out of 120 people will get the flu this winter.

Step-by-step explanation:

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And p is the probability of X happening.

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 , .

Assume the probability that a given person will get the flu this winter is 97%.

This means that

120 people

This means that

Verifying the necessary conditions.

Since , the normal distribution cannot be used as an approximation to the binomial probability to approximate the probability. Thus, the binomial distribution has to be used.

Probability using the binomial distribution:

Probability close to 0, but below the mean, which means that the probability of the number being above this is 100%.

Using the binomial distribution, 100% probability that more than 97 out of 120 people will get the flu this winter.

step-by-step explanation:

try remembering the steps

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