High School

Regular gas prices (in 2011\$) for a sample of 5 Iowa stations in September 2011 were: 3.66, 3.63, 3.72, 3.60, and 3.58. The buying power equation, \(130.74(2011\$) = 170.57 (2022\$)\), was derived using the BLS CPI Inflation Calculator.

2(a) What was the average price (in 2011\$) of regular gas in September 2011 for the sample of 5 Iowa stations?

2(b) What was the standard deviation of regular gas prices (in 2011\$) in September 2011 for the sample of 5 Iowa stations?

2(c) What was the 90th percentile of regular gas prices (in 2011\$) in September 2011 for the sample of 5 Iowa stations?

2(d) What was the average inflation-adjusted (to 2022\$) price of regular gas in September 2011 for the sample of 5 Iowa stations? (Hint: Each of the 5 adjusted prices will be based on the buying-power equation.)

2(e) The average price of regular gas in Iowa on July 27, 2022, was $4.00. Compare the price of gas in 2022 (\$4.00) to the average inflation-adjusted (to 2022\$) price of gas in 2011 from part 2(d). Specifically, give the inflation-adjusted percent change in regular gas price from September 2011 to July 2022; that is, compute:

\[
\left(\frac{\text{avg(2011 adjusted prices)} - \text{2022 price}}{\text{avg(2011 adjusted prices)}}\right) \times 100\%
\]

Answer :

2(a) The average price of regular gas in September 2011 for the sample of 5 Iowa stations was approximately $3.638 in 2011$.

2(b) The standard deviation of regular gas prices in September 2011 for the sample of 5 Iowa stations was approximately $0.053 in 2011$.

2(c) The 90th percentile of regular gas prices in September 2011 for the sample of 5 Iowa stations was approximately $3.714 in 2011$.

2(d) The average inflation-adjusted price of regular gas in September 2011 for the sample of 5 Iowa stations was approximately $4.757 in 2022$.

2(e) The inflation-adjusted percent change in regular gas price from September 2011 to July 2022 is approximately -15.91%.

To solve these questions, we'll go step by step:

2(a) Average price (in 2011$) of regular gas in September 2011 for the sample of 5 Iowa stations:

To find the average, we add up all the prices and divide by the number of stations:

Average price = (3.66 + 3.63 + 3.72 + 3.60 + 3.58) / 5

= 18.19 / 5

= 3.638 (rounded to three decimal places)

Therefore, the average price of regular gas in September 2011 for the sample of 5 Iowa stations was approximately $3.638 in 2011$.

2(b) Standard deviation of regular gas prices (in 2011$) in September 2011 for the sample of 5 Iowa stations:

To calculate the standard deviation, we need to find the deviation of each price from the average, square each deviation, find the average of the squared deviations, and finally take the square root.

Deviation from the mean: (3.66 - 3.638), (3.63 - 3.638), (3.72 - 3.638), (3.60 - 3.638), (3.58 - 3.638)

= 0.022, -0.008, 0.082, -0.038, -0.058

Squared deviations: 0.000484, 0.000064, 0.006724, 0.001444, 0.003364

Average of squared deviations = (0.000484 + 0.000064 + 0.006724 + 0.001444 + 0.003364) / 5

= 0.0028188

Standard deviation = √0.0028188

≈ 0.053 (rounded to three decimal places)

Therefore, the standard deviation of regular gas prices in September 2011 for the sample of 5 Iowa stations was approximately $0.053 in 2011$.

2(c) 90th percentile of regular gas prices (in 2011$) in September 2011 for the sample of 5 Iowa stations:

To find the 90th percentile, we arrange the prices in ascending order and select the value that corresponds to the 90th percentile.

Arranged prices: 3.58, 3.60, 3.63, 3.66, 3.72

The 90th percentile falls between the 4th and 5th prices. We can interpolate to find the exact value.

90th percentile = 3.66 + (0.9 * (3.72 - 3.66))

= 3.66 + (0.9 * 0.06)

= 3.66 + 0.054

= 3.714 (rounded to three decimal places)

Therefore, the 90th percentile of regular gas prices in September 2011 for the sample of 5 Iowa stations was approximately $3.714 in 2011$.

2(d) Average inflation-adjusted (to 2022$) price of regular gas in September 2011 for the sample of 5 Iowa stations:

We'll use the given buying power equation to adjust the prices from 2011$ to 2022$. Each price will be multiplied by the ratio of 170.57 (2022$) to 130.74 (2011$).

Adjusted prices = (3.66 * (170.57 / 130.74)), (3.63 * (170.57 / 130.74)), (3.72 * (170.57 / 130.74)), (3.60 * (170.57 / 130.74)), (3.58 * (170.57 / 130.74))

Calculating each adjusted price and rounding to three decimal places:

Adjusted prices = 4.775, 4.741, 4.863, 4.711, 4.693

Average adjusted price = (4.775 + 4.741 + 4.863 + 4.711 + 4.693) / 5

= 23.783 / 5

= 4.757 (rounded to three decimal places)

Therefore, the average inflation-adjusted price of regular gas in September 2011 for the sample of 5 Iowa stations was approximately $4.757 in 2022$.

2(e) Inflation-adjusted percent change in regular gas price from September 2011 to July 2022:

To calculate the inflation-adjusted percent change, we'll compare the average inflation-adjusted price of gas in 2011 to the price of gas in 2022.

Inflation-adjusted percent change = ((4.00 - 4.757) / 4.757) * 100%

Calculating the inflation-adjusted percent change:

Inflation-adjusted percent change = ((4.00 - 4.757) / 4.757) * 100%

= (-0.757 / 4.757) * 100%

≈ -15.91%

Therefore, the inflation-adjusted percent change in regular gas price from September 2011 to July 2022 is approximately -15.91%.

Learn more about standard deviation here:

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