The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough samples of any data, the means of the samples can be modelled by a normal distribution. Psychologists use the normal distribution to classify people's test results, including intelligence tests. Don't be a dummy in your IB exam and make sure you understand this topic!
Key Concepts
On this page, you should learn about
the normal distribution and its curve
probability calculations with the normal distribution
standardization of the normal variables
finding mean and/or variance using inverse normal calculations
The weights of female adult penguins found on a remote island are normally distributed with a mean of 3.15 kg. It is found that 81% of these penguins weigh more than 3 kg. The weights of the penguins have a standard deviation of \(\sigma\).
a) Find the standardized value for 3kg.
b) Hence, find the value of \(\sigma\).
It is estimated that there are 4000 female adult penguins on the island. Penguins are considered underweight if they weigh less than 2.9 kg.
c) Find the estimated number of underweight female penguins on the island.
Hint
Since you use the answers from one part of the question oto the next, it is important to keep a high degree of accuracy in your answers.
a) Use InvNorm
b) 'Hence' means that this should follow on from the previous part. Use your z value to find \(\sigma\) using standardizing formula \(Z=\frac{X- \mu }{\sigma}\)
It is known that 2% of Filips lightbulbs have a life of less than 1000 hours and 10% have a life less than 1100 hours. It can be assumed that lightbulb life is normally distributed with a mean of \(\mu\) and a standard deviation of \(\sigma\)
a) Find the value of \(\mu\) and the value of \(\sigma\).
b) Find the probability that a randomly selected Filips lightbulb will have a life of at least 1200 hours.
Hint
a) In this question we need to standardize the normal distribution.
Use InvNorm to find the corresponding z values for 1000 hours and 1100 hours.
Solve 2 simultaneous equations to find \(\mu\) and \(\sigma\).