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A recent survey found that about 72.5% of all gasoline purchases at a certain service station chain are paid with a credit or debit card. If 300 gasoline purchases completed at this chain are randomly selected, find the probability that at most 210 of those purchases are paid with a credit or debit card.

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Answer:

18.28% probability that at most 210 of those purchases are paid with a credit or debit card.

Explanation:

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:


√(V(X)) = √(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 = (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 = √(V(X)).

In this problem, we have that:


n = 300, p = 0.725

So


\mu = E(X) = np = 300*0.725 = 217.5


\sigma = √(V(X)) = √(np(1-p)) = √(300*0.725*0.275) = 7.7339

Find the probability that at most 210 of those purchases are paid with a credit or debit card.

Using continuity correction, this is
P(X \leq 210 + 0.5) = P(X \leq 210.5), which is 1 subtracted by the pvalue of Z when X = 210.5. So


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


Z = (210.5 - 217.5)/(7.7339)


Z = -0.905


Z = -0.905 has a pvalue of 0.1828

18.28% probability that at most 210 of those purchases are paid with a credit or debit card.

User Kirti Nikam
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