Need help! A normal distribution with a mean of 20 and a standard deviation of 2 is evaluated to solve a business problem. What is the probability that the value in the population is between 20 and 30? What is the probability that any value in the population is between 10 and 30?
Thank you!
This is a great little stats calculator, just enter your values
http://davidmlane.com/hyperstat/z_table.html
To find the probability that a value in a normal distribution falls within a certain range, you need to calculate the area under the normal curve that corresponds to that range. This can be done using the standard normal distribution table or by using a statistical software or calculator.
1. Probability that the value is between 20 and 30:
To calculate this probability, you need to calculate the area under the normal curve between the z-scores corresponding to 20 and 30. First, you need to convert the values of 20 and 30 into z-scores using the formula:
z = (x - mean) / standard deviation
For the value 20:
z1 = (20 - 20) / 2 = 0
For the value 30:
z2 = (30 - 20) / 2 = 5
Since the z-table provides probabilities for values up to 3.49, the probability for z2 = 5 would be essentially 1.
Therefore, the probability that a value in the population is between 20 and 30 is approximately 1.
2. Probability that any value in the population is between 10 and 30:
To calculate this probability, you need to calculate the area under the normal curve between the z-scores corresponding to 10 and 30.
For the value 10:
z1 = (10 - 20) / 2 = -5
For the value 30:
z2 = (30 - 20) / 2 = 5
Looking up the z-scores in the z-table, you'll find that the area to the left of z = -5 is extremely close to 0 and the area to the left of z = 5 is essentially 1. Therefore, the probability that any value in the population is between 10 and 30 is approximately 1.