Question

In: Math

What are two examples for normal distributions with numbers and formula?

What are two examples for normal distributions with numbers and formula?

Solutions

Expert Solution

# Example of normal distribution are as follows

Example 1

An average life of light bulb manufactured by company will 300 days with standard deviation of 50 day . Assuming that bulb life is normally distributed  what is the probability that an life of light bulb will at most 365 days?

Ans: Given a mean score of 300 days and a standard deviation of 50 days, we want to find the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the following:

  • The value of the normal random variable is 365 days.
  • The mean is equal to 300 days.
  • The standard deviation is equal to 50 days.

We enter these values into the Normal Distribution Calculator and compute the cumulative probability. The answer is: P( X < 365) = 0.90. Hence, there is a 90% chance that a light bulb will burn out within 365 days.

Example 2

Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110?

Solution: Here, we want to know the probability that the test score falls between 90 and 110. The "trick" to solving this problem is to realize the following:

P( 90 < X < 110 ) = P( X < 110 ) - P( X < 90 )

We use the Normal Distribution Calculator to compute both probabilities on the right side of the above equation.

  • To compute P( X < 110 ), we enter the following inputs into the calculator: The value of the normal random variable is 110, the mean is 100, and the standard deviation is 10. We find that P( X < 110 ) is 0.84.
  • To compute P( X < 90 ), we enter the following inputs into the calculator: The value of the normal random variable is 90, the mean is 100, and the standard deviation is 10. We find that P( X < 90 ) is 0.16.

We use these findings to compute our final answer as follows:

P( 90 < X < 110 ) = P( X < 110 ) - P( X < 90 )
P( 90 < X < 110 ) = 0.84 - 0.16
P( 90 < X < 110 ) = 0.68

Thus, about 68% of the test scores will fall between 90 and 110.


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