Question

The Normal Probability distribution has many practical uses. Please provide some examples of real life data sets that are normally distributed.

Answer 1

Example 1:

A set of data is normally distributed with a mean of 55 . What percent of the data is less than 55 ?

A normal distribution is symmetric about the mean. So, half of the data will be less than the mean and half of the data will be greater than the mean.

Therefore, 50%percent of the data is less than 55 .

Example 2:

The life of a fully-charged cell phone battery is normally distributed with a mean of 14 hours with a standard deviation of 11hour. What is the probability that a battery lasts at least 13 hours?

The mean is 14 and the standard deviation is 1 .

50% of the normal distribution lies to the right of the mean, so 50% of the time, the battery will last longer than 14 hours.

The interval from 13 to 14 hours represents one standard deviation to the left of the mean. So, about 34% of time, the battery will last between 13 and 14hours.

Therefore, the probability that the battery lasts at least 13 hours is about34%+50% or 0.84 .

Example 3:

The average weight of a raspberry is 4.4gm with a standard deviation of 1.3 gm. What is the probability that a randomly selected raspberry would weigh at least 3.1gm but not more than 7.0 gm?

The mean is 4.4 and the standard deviation is 1.3 .

Note that

4.4−1.3=3.1

and

4.4+2(1.3)=7.04

So, the interval 3.1≤x≤7.0is actually between one standard deviation below the mean and 2 standard deviations above the mean.

In normally distributed data, about 34%of the values lie between the mean and one standard deviation below the mean, and34% between the mean and one standard deviation above the mean.

In addition, 13.5%of the values lie between the first and second standard deviations above the mean.

Adding the areas, we get34%+34%+13.5%=81.5%

Therefore, the probability that a randomly selected raspberry will weigh at least 3.1gm but not more than 7.0 gm is 81.5% or0.815