Probability can be defined as the likelihood of an event to occur. In statistics, the mean of the sample distribution typically shows the probability of the population.

From the parameters given:

The sample size (n) = 55 patients
Let's assume that the mean (x) = 32 (i.e. 58.2%) of the patients

The sample proportion $\mathbf{\hat p}$ can be computed by using the expression:

$\mathbf{\hat p = \dfrac{x}{n}}$

$\mathbf{\hat p = \dfrac{32}{55}}$

$\mathbf{\hat p =0.582}$

If the percentage of the probability of all patients in the emergency room = 0.75

The probability that the proportion of patients who wait less than 30 minutes is 0.582 or less can be computed as:

$\mathbf{( \hat P \leq 0.582) = P \Big( \dfrac{\hat p - p}{\sqrt{\dfrac{p(1-p)}{n}}} \leq \dfrac{0.582 - 0.75}{\sqrt{\dfrac{0.75(1-0.75)}{55}}} \Big )}$

$\mathbf{( \hat P \leq 0.582 )= P \Big( Z \leq \dfrac{-0.168}{\sqrt{0.003409}} \Big )}$

$\mathbf{( \hat P \leq 0.582) = P \Big( Z \leq -2.88 \Big )}$

From the Z distribution table:

$\mathbf{( \hat P \leq 0.582) = 0.00198}$

$\mathbf{( \hat P \leq 0.582) \simeq 0.0020}$

Learn more about probability here:

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