The average age of doctors in certain hospital is 43 years old. Suppose distribution of ages is normal and the standard deviation is 8.0 years. If 25 doctors is chosen at random for a committee, find the probability the average age for those doctors is less than 43.8 years. Assume the variable is normally distributed.
Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 210 feet and a standard deviation of 44 feet. Let X = distance in feet for a fly ball.
Z = (score-mean)/SEm
SEm = SD/√n
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.
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To find the probability of the average age for those doctors being less than 43.8 years, we can use the Central Limit Theorem. According to the Central Limit Theorem, if you take repeated random samples of a certain size from any population with a mean and a standard deviation, the distribution of the sample means will approach a normal distribution as the sample size increases.
In this case, the population is the age distribution of doctors in the hospital, with a mean of 43 years and a standard deviation of 8 years. We are interested in the average age of a sample of 25 doctors.
To calculate the probability, we need to convert the sample mean to a z-score. The z-score tells us how many standard deviations the sample mean is from the population mean. It is calculated using the formula:
z = (x - μ) / (σ / √n)
Where:
- x is the sample mean (43.8 years)
- μ is the population mean (43 years)
- σ is the population standard deviation (8 years)
- n is the sample size (25 doctors)
- √ denotes square root
Let's calculate the z-score:
z = (43.8 - 43) / (8 / √25)
= 0.8 / (8 / 5)
= 0.8 / 1.6
= 0.5
Now that we have the z-score, we can use a standard normal distribution table or a calculator to find the corresponding probability. The probability can be found by looking up the z-score in the table or using the calculator.
For example, using a standard normal distribution table, you can find that the probability associated with a z-score of 0.5 is approximately 0.6915.
Therefore, the probability that the average age for those doctors is less than 43.8 years is approximately 0.6915 or 69.15%.