Multiple Choice
According to the law of large numbers, as the sample size increases, which of the following statements is true about the ?
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7. Sampling Distributions & Confidence Intervals: Mean
Introduction to Confidence Intervals
Multiple Choice
Suppose the waiting times for patients needing emergency service are normally distributed with a mean of minutes and a standard deviation of minutes. What proportion of patients wait minutes or less?
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Verified step by step guidance1
Identify the given parameters: the waiting times are normally distributed with a mean \( \mu = 12 \) minutes and a standard deviation \( \sigma = 3 \) minutes.
Define the problem: we want to find the proportion of patients who wait 9 minutes or less. This means we are looking for \( P(X \leq 9) \), where \( X \) is the waiting time.
Convert the raw score (9 minutes) to a standard normal score (z-score) using the formula:
\[ z = \frac{X - \mu}{\sigma} \]
Substitute the values to get:
\[ z = \frac{9 - 12}{3} \]
Use the standard normal distribution table (z-table) or a calculator to find the cumulative probability corresponding to the calculated z-score. This gives the proportion of patients waiting 9 minutes or less.
Interpret the result: the value obtained from the z-table is the proportion of patients whose waiting time is at most 9 minutes.
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