Question

The average annual inflation rate in the United States over the past 98 years is 3.37% and has a standard deviation of approximately 5% (Inflationdata). In 1980, the inflation rate was above 13%. If the annual inflation rate is normally distributed, what is the probability that inflation will be above 13% next year

Answer 1

`P(X>13)=P((X-\mu)/(\sigma)>(13-\mu)/(\sigma))=P(Z>(13-3.37)/(5))=P(Z>1.926)`

And we can find this probability using the complement rule and the normal standard distirbution table or excel:

`P(Z>1.926)=1-P(Z<1.926)=1-0.9729=0.0271`

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 annual inflation of a population, and for this case we know the distribution for X is given by:

`X \sim N(3.37,5)`

Where
`\mu=3.37` and
`\sigma=5`

We are interested on this probability

`P(X>13)`

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>13)=P((X-\mu)/(\sigma)>(13-\mu)/(\sigma))=P(Z>(13-3.37)/(5))=P(Z>1.926)`

And we can find this probability using the complement rule and the normal standard distirbution table or excel:

`P(Z>1.926)=1-P(Z<1.926)=1-0.9729=0.0271`

Answer 2

2.68% probability that inflation will be above 13% next year

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 3.37, \sigma = 5`

If the annual inflation rate is normally distributed, what is the probability that inflation will be above 13% next year

This is the pvalue of Z when X = 13. So

`Z = (X - \mu)/(\sigma)`

`Z = (13 - 3.37)/(5)`

`Z = 1.93`

`Z = 1.93` has a pvalue of 0.9732

1 - 0.9732 = 0.0268

2.68% probability that inflation will be above 13% next year