Question

Todd Holland from NY grows pumpkins. He studied his past records carefully and concluded that his biggest pumpkins are distributed according to a normal distribution with mean 950 lbs and standard deviation 50 lbs. (a) What is the probability for Todd to get his biggest pumpkin weighing more than 1000 lbs

Answer 1

`P(X>1000)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X>1000)=P((X-\mu)/(\sigma)>(1000-\mu)/(\sigma))=P(Z>(1000-950)/(50))=P(z>1)`

And we can find this probability using the complement rule and the normal standard table and we got:

`P(z>1)=1-P(z<1)=1-0.841=0.159`

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 problem

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

`X \sim N(950,50)`

Where
`\mu=950` and
`\sigma=50`

We are interested on this probability

`P(X>1000)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X>1000)=P((X-\mu)/(\sigma)>(1000-\mu)/(\sigma))=P(Z>(1000-950)/(50))=P(z>1)`

And we can find this probability using the complement rule and the normal standard table and we got:

`P(z>1)=1-P(z<1)=1-0.841=0.159`