Question

The duration of routine operations at charlotte general has approximately a normal distribution with an average of 125 minutes and a standard deviation of 18 minutes.

a.what proportion of operations last at least 120 minutes?
b.what proportion of operations lasted between 90 and 130 minutes?
c.what proportion of operations lasted less than 122 minutes?

Answer 1

To find the proportion of operations that last at least 120 minutes, convert the distribution to a standard one and find the area to the right of 120 minutes. The proportion is approximately 0.3907 or 39.07%. To find the proportion of operations between 90 and 130 minutes, find the area between the two values using the standard distribution. The proportion is approximately 0.7910 or 79.10%. To find the proportion of operations lasting less than 122 minutes, find the area to the left of 122 minutes using the standard distribution. The proportion is approximately 0.4332 or 43.32%.

Step-by-step explanation:

To solve this problem, we can use the concept of standard normal distribution. In order to find the proportion of operations that last at least 120 minutes, we need to find the area under the normal distribution curve to the right of 120 minutes. We can convert the original distribution to a standard normal distribution by using the z-score formula: z = (x - μ) / σ. Plugging in the values for our problem, we have z = (120 - 125) / 18 = -0.278. Using a z-table or calculator, we can find that the proportion of operations that last at least 120 minutes is approximately 0.3907, or 39.07%.

To find the proportion of operations lasting between 90 and 130 minutes, we can find the area under the normal distribution curve between these two values. We can again convert the original distribution to a standard normal distribution by using the z-score formula. Plugging in the values for our problem, we have z1 = (90 - 125) / 18 = -1.944 and z2 = (130 - 125) / 18 = 0.278. Using a z-table or calculator, we can find that the proportion of operations lasting between 90 and 130 minutes is approximately 0.7910, or 79.10%.

To find the proportion of operations lasting less than 122 minutes, we can find the area under the normal distribution curve to the left of 122 minutes. Again, we can convert the original distribution to a standard normal distribution by using the z-score formula. Plugging in the values for our problem, we have z = (122 - 125) / 18 = -0.167. Using a z-table or calculator, we can find that the proportion of operations lasting less than 122 minutes is approximately 0.4332, or 43.32%.

Answer 2

To calculate the proportion of operations lasting a certain amount of time at Charlotte General, we use the properties of the normal distribution to convert operation times to Z-scores and then use these Z-scores to find the required proportions.

Step-by-step explanation:

To solve these problems involving the normal distribution, we will use the properties of a normally distributed variable and standard normal distribution tables (Z-tables) or a calculator with normal distribution functions. Since the questions are based on the duration of operations at Charlotte General, we will first convert the operation times into Z-scores and then find the proportion of operations that fall within the specified times using the Z-scores.

• a. Proportion of operations lasting at least 120 minutes: To find this, calculate the Z-score for 120 minutes using the formula Z = (X - μ) / σ, where X is 120 minutes, μ (mean) is 125 minutes, and σ (standard deviation) is 18 minutes. Then, look up the Z-score in a standard normal distribution table or use a calculator to find the proportion of values to the right of this Z-score, which represents operations lasting at least 120 minutes.
• b. Proportion of operations lasting between 90 and 130 minutes: Calculate the Z-scores for both 90 and 130 minutes, then find the area between these two Z-scores using a standard normal distribution table or a calculator.
• c. Proportion of operations lasting less than 122 minutes: Calculate the Z-score for 122 minutes and find the area to the left of this Z-score using a standard normal distribution table or a calculator, representing the operations lasting less than 122 minutes.