1- The ages of a population of 300 is normally distributed, with a mean of 45 and a standard deviation of 4. If you randomly selected a member of the population, what is the probablity that his or her age would be greater than 49?
To find the probability that a randomly selected member of the population would have an age greater than 49, we can use the standard normal distribution.
First, we need to calculate the z-score for 49. The z-score measures the number of standard deviations an observation is from the mean. It is calculated using the formula:
z = (x - μ) / σ
Where:
- x is the value we want to find the probability for (in this case, 49)
- μ is the mean of the distribution (45)
- σ is the standard deviation of the distribution (4)
Plugging in the values, we get:
z = (49 - 45) / 4
z = 4 / 4
z = 1
Now, we can use the standard normal distribution table (also known as the Z-table) or a calculator to find the probability corresponding to this z-score. The table or calculator will give us the area under the standard normal curve from the z-score of 1 to positive infinity.
Using the Z-table, we can look up the area corresponding to a z-score of 1. This value represents the probability of a randomly selected member having an age less than or equal to 49. To find the probability of having an age greater than 49, we need to subtract this value from 1.
If we look up the area corresponding to a z-score of 1 in the Z-table, we find that it is approximately 0.8413. Therefore, the probability of a randomly selected member having an age greater than 49 is:
1 - 0.8413 ≈ 0.1587
So, the probability is approximately 0.1587, or 15.87%.