Question

A major television manufacturer has determined that its 44 inch screens have a mean service life that can be modeled by a normal distribution with a mean of 6 years and a standard deviation of one-half year (6 months). What is the probability that the service life of that product is between 5 and 7 years

Answer 1

Answer:

$P(5$

And we can find this probability with thie difference:

$P(-2$

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

$P(-2$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

$X \sim N(6,0.5)$  

Where $\mu=6$ and $\sigma=0.5$

We are interested on this probability

$P(5$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(5$

And we can find this probability with thie difference:

$P(-2$

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

$P(-2$