At one point the average price of regular unleaded gasoline was $3.41 per gallon. Assume that the standard deviation price per gallon is $0.09 per gallon and use Chebyshev's inequality to answer the following.

(a) What percentage of gasoline stations had prices within 2 standard deviations of the mean?
(b) What percentage of gasoline stations had prices within 1.5 standard deviations of the mean? What are the gasoline prices that are within 1.5 standard deviations of the mean?
(c) What is the minimum percentage of gasoline stations that had prices between $3.05 and $3.77?

a)  

So we expected about 75% within two deviations from the mean

b)

So we expected about 55.6% within 1.5 deviations from the mean

And the limits are:

c) We can calculate how many deviations we are within the mean with the limits with this formula:

And using the lower limit we got:

And with the upper limit we got:

So then the value of k =4 and the percentage is given by:

Step-by-step explanation:

Previous concepts and Data given  

reprsent the population mean

represent the population standard deviation

The Chebyshev's Theorem states that for any dataset

•We have at least 75% of all the data within two deviations from the mean.

•We have at least 88.9% of all the data within three deviations from the mean.

•We have at least 93.8% of all the data within four deviations from the mean.

Or in general words "For any set of data (either population or sample) and for any constant k greater than 1, the proportion of the data that must lie within k standard deviations on either side of the mean is at least:

Part a

For this case we can find the percentage required replaincg k =2 and we got:

So we expected about 75% within two deviations from the mean

Part b

For this case we can find the percentage required replaincg k =2 and we got:

So we expected about 55.6% within 1.5 deviations from the mean

And the limits are:

Part c

We can calculate how many deviations we are within the mean with the limits with this formula:

And using the lower limit we got:

And with the upper limit we got:

So then the value of k =4 and the percentage is given by: