In December 2010, the average price of regular unleaded gasoline excluding taxes in the United States was $3.06 per gallon, according to the Energy Information Administration. Assume that the standard deviation price per gallon is $0.06 per gallon to answer the following.

(a) What minimum percentage of gasoline stations had prices within 3 standard deviations of the mean?

(b) What minimum 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 $2.94 and $3.18?

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(a) To find the minimum percentage of gasoline stations that had prices within 3 standard deviations of the mean, we can use the empirical rule, which states that for a normal distribution:

- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Since we want to find the minimum percentage, we can assume that the distribution is normal and use the empirical rule. Therefore, approximately 99.7% of gasoline stations had prices within 3 standard deviations of the mean.

(b) To find the minimum percentage of gasoline stations that had prices within 2.5 standard deviations of the mean, we can use the same approach as in part (a). Since 2.5 is between 2 and 3, we can estimate that the percentage is still around 99.7%.

To find the prices that are within 2.5 standard deviations of the mean, we can use the following formula:
Lower Bound = Mean - (2.5 x Standard Deviation)
Upper Bound = Mean + (2.5 x Standard Deviation)

Substituting the given values:
Lower Bound = $3.06 - (2.5 x $0.06) = $3.06 - $0.15 = $2.91
Upper Bound = $3.06 + (2.5 x $0.06) = $3.06 + $0.15 = $3.21

Therefore, the gasoline prices that are within 2.5 standard deviations of the mean range from $2.91 to $3.21 per gallon.

(c) To find the minimum percentage of gasoline stations that had prices between $2.94 and $3.18, we can use the cumulative probability function for a normal distribution. First, we need to standardize the prices using the z-score formula:

z = (X - Mean) / Standard Deviation

For the lower bound:
z_lower = ($2.94 - $3.06) / $0.06 = -0.2 / $0.06 = -3.33

For the upper bound:
z_upper = ($3.18 - $3.06) / $0.06 = 0.12 / $0.06 = 2.00

Next, we can use a z-table or a calculator to find the cumulative probability associated with these z-scores. Subtracting the lower cumulative probability from the upper cumulative probability will give us the minimum percentage.

Using a z-table or a calculator, we find:
- The cumulative probability for z = -3.33 is approximately 0.0004.
- The cumulative probability for z = 2.00 is approximately 0.9772.

Therefore, the minimum percentage of gasoline stations that had prices between $2.94 and $3.18 is approximately 97.72% - 0.04% = 97.68%.

To answer these questions, we need to understand the concept of standard deviations and how they relate to the normal distribution. The normal distribution represents a bell-shaped curve that is symmetric around the mean. The standard deviation measures the average distance of data points from the mean.

(a) To find the minimum percentage of gasoline stations with prices within 3 standard deviations of the mean, we can use the empirical rule or the 68-95-99.7 rule. According to this rule, approximately 99.7% of the data lies within 3 standard deviations of the mean in a normal distribution. However, in this case, we have to exclude the prices below zero (since prices cannot be negative). So, we need to find the percentage of data that falls within three standard deviations above zero, which will be slightly lower than 99.7%.

To calculate this value, we can subtract the cumulative percentage of data outside three standard deviations (above or below) from 100%. Since the normal distribution is symmetric, we can calculate the percentage above (or below) three standard deviations and double it.

Using the cumulative distribution function (CDF) of the normal distribution, we can find the percentage by calculating:

P(x <= 3σ) = P(x <= μ + 3σ) = CDF(3σ)

To calculate this value, we need to transform the given information into a standard normal distribution, where the mean is 0 and the standard deviation is 1. To do this, we subtract the mean from every value and divide by the standard deviation:

Z = (x - μ) / σ

So, for this case, the standard deviation (σ) is $0.06 per gallon, and the mean (μ) is $3.06 per gallon.

Z = (x - 3.06) / 0.06

Using a standard normal distribution table or a calculator, we can find CDF(3) = 0.9987, which is the percentage for one tail. To get the percentage for both tails, we subtract it from 1 and then multiply by 2:

P(x <= 3σ) = (1 - 0.9987) * 2

Calculating this, we find:

P(x <= 3σ) ≈ 0.0039 or 0.39%

So, the minimum percentage of gasoline stations that had prices within 3 standard deviations of the mean is approximately 0.39%.

(b) Similarly, to find the minimum percentage of gasoline stations with prices within 2.5 standard deviations of the mean, we can use the same approach. However, in this case, we need to calculate the gasoline prices that fall within 2.5 standard deviations of the mean.

Using the same formula as before:

Z = (x - 3.06) / 0.06

We find that a Z-value of 2.5 corresponds to (2.5 * 0.06) + 3.06 = $3.21 per gallon. This is the upper limit.

P(x <= 2.5σ) = P(x <= μ + 2.5σ) = CDF(2.5σ)

Using a standard normal distribution table or a calculator, we can find CDF(2.5) = 0.9938. To get both tails:

P(x <= 2.5σ) = (1 - 0.9938) * 2

Calculating this, we find:

P(x <= 2.5σ) ≈ 0.0124 or 1.24%

So, the minimum percentage of gasoline stations that had prices within 2.5 standard deviations of the mean is approximately 1.24%. The gasoline prices that are within 2.5 standard deviations of the mean are between $3.06 and $3.21 per gallon.

(c) To find the minimum percentage of gasoline stations with prices between $2.94 and $3.18, we need to convert these prices to Z-scores (standardize them) based on the given mean and standard deviation.

For $2.94 per gallon:
Z1 = ($2.94 - $3.06) / $0.06

For $3.18 per gallon:
Z2 = ($3.18 - $3.06) / $0.06

Using the same approach, we can calculate the percentage between Z1 and Z2:

P(Z1 <= Z <= Z2) = CDF(Z2) - CDF(Z1)

Using a standard normal distribution table or a calculator, we can find the CDF values for Z1 and Z2 and calculate the difference to find the percentage between $2.94 and $3.18.