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Complete Question
At one point the average price of regular unleaded gasoline was $3.39 per gallon. Assume that the standard deviation price per gallon is $0.07 per gallon and use Chebyshev's inequality to answer the following.
(a) What percentage of gasoline stations had prices within 3 standard deviations of the mean?
(b) What percentage of gasoline stations had prices within 2.5 standard deviations of the mean? What are the gasoline prices that are within 2.5 standard deviations of the mean?
(c) What is the minimum percentage of gasoline stations that had prices between $3.11 and $3.67?
Answer:
a) 88.89% lies with 3 standard deviations of the mean
b) i) 84% lies within 2.5 standard deviations of the mean
ii) the gasoline prices that are within 2.5 standard deviations of the mean is $3.215 and $3.565
c) 93.75%
Step-by-step explanation:
Chebyshev's theorem is shown below.
1) Chebyshev's theorem states for any k > 1, at least 1-1/k² of the data lies within k standard deviations of the mean.
As stated, the value of k must be greater than 1.
2) At least 75% or 3/4 of the data for a set of numbers lies within 2 standard deviations of the mean. The number could be greater.μ - 2σ and μ + 2σ.
3) At least 88.89% or 8/9 of a data set lies within 3 standard deviations of the mean.μ - 3σ and μ + 3σ.
4) At least 93.75% of a data set lies within 4 standard deviations of the mean.μ - 4σ and μ + 4σ.
(a) What percentage of gasoline stations had prices within 3 standard deviations of the mean?
We solve using the first rule of the theorem
1) Chebyshev's theorem states for any k > 1, at least 1-1/k² of the data lies within k standard deviations of the mean.
As stated, the value of k must be greater than 1.
Hence, k = 3
1 - 1/k²
= 1 - 1/3²
= 1 - 1/9
= 9 - 1/ 9
= 8/9
Therefore, the percentage of gasoline stations had prices within 3 standard deviations of the mean is 88.89%
(b) What percentage of gasoline stations had prices within 2.5 standard deviations of the mean?
We solve using the first rule of the theorem
1) Chebyshev's theorem states for any k > 1, at least 1-1/k² of the data lies within k standard deviations of the mean.
As stated, the value of k must be greater than 1.
Hence, k = 3
1 - 1/k²
= 1 - 1/2.5²
= 1 - 1/6.25
= 6.25 - 1/ 6.25
= 5.25/6.25
We convert to percentage
= 5.25/6.25 × 100%
= 0.84 × 100%
= 84 %
Therefore, the percentage of gasoline stations had prices within 2.5 standard deviations of the mean is 84%
What are the gasoline prices that are within 2.5 standard deviations of the mean?
We have from the question, the mean =$3.39
Standard deviation = 0.07
μ - 2.5σ
$3.39 - 2.5 × 0.07
= $3.215
μ + 2.5σ
$3.39 + 2.5 × 0.07
= $3.565
Therefore, the gasoline prices that are within 2.5 standard deviations of the mean is $3.215 and $3.565
(c) What is the minimum percentage of gasoline stations that had prices between $3.11 and $3.67?
the mean =$3.39
Standard deviation = 0.07
Applying the 2nd rule
2) At least 75% or 3/4 of the data for a set of numbers lies within 2 standard deviations of the mean. The number could be greater.μ - 2σ and μ + 2σ.
the mean =$3.39
Standard deviation = 0.07
μ - 2σ and μ + 2σ.
$3.39 - 2 × 0.07 = $3.25
$3.39 + 2× 0.07 = $3.53
Applying the third rule
3) At least 88.89% or 8/9 of a data set lies within 3 standard deviations of the mean.μ - 3σ and μ + 3σ.
$3.39 - 3 × 0.07 = $3.18
$3.39 + 3 × 0.07 = $3.6
Applying the 4th rule
4) At least 93.75% of a data set lies within 4 standard deviations of the mean.μ - 4σ and μ + 4σ.
$3.39 - 4 × 0.07 = $3.11
$3.39 + 4 × 0.07 = $3.67
Therefore, from the above calculation we can see that the minimum percentage of gasoline stations that had prices between $3.11 and $3.67 corresponds to at least 93.75% of a data set because it lies within 4 standard deviations of the mean.