mean=20, standard deviation =6.Use the emperical rule to estimate the probabilty that a randomly chosen patient had to wait more than 26 min. Of 2000 patients, estimate the # of patient having to wait more than 26 mins
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply that probability by 2000 to get your second answer.
To estimate the probability and the number of patients who had to wait more than 26 minutes using the empirical rule, we need to understand the concept of the empirical rule and apply it to the given mean and standard deviation.
The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule that applies to data that follows a normal distribution. According to the empirical rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Given that the mean is 20 and the standard deviation is 6, we can apply the empirical rule to estimate the probability and the number of patients who had to wait more than 26 minutes.
Step 1: Calculate the z-score:
The z-score measures how many standard deviations an observation is from the mean. We can calculate the z-score using the formula:
z = (x - mean) / standard deviation
Here, x represents the value we want to calculate the probability for, which is 26 minutes.
z = (26 - 20) / 6
z = 1
Step 2: Determine the probability:
Since we are looking for the probability that a randomly chosen patient had to wait more than 26 minutes, we need to find the area under the normal distribution curve to the right of the z-score of 1.
Using a standard normal distribution table or a calculator, we find that the probability corresponding to a z-score of 1 is approximately 0.1587. This means that the probability of a patient having to wait more than 26 minutes is 0.1587.
Step 3: Estimate the number of patients:
To estimate the number of patients who had to wait more than 26 minutes out of 2000 patients, we can multiply the probability by the total number of patients.
Estimated # of patients = Probability * Total # of patients
Estimated # of patients = 0.1587 * 2000
Estimated # of patients ≈ 317.4
Therefore, we can estimate that around 317 patients out of the 2000 patients had to wait more than 26 minutes.