Question

Suppose you are told that a 95% confidence interval for the average price of a gallon of regular gasoline in your state is from $3.04 to $3.61. Use the fact that the confidence interval for the mean is in the form x − E to x + E to compute the sample mean and the maximal margin of error E. (Round your answers to two decimal places.)

Answer 1

Answer 2

Given:

• 95% Confidence Interval (CI) = $3.04to$3.61.

• CI is in the form:

  $$\text{CI}=(\bar{x}-E,\bar{x}+E)$$

  where:

• $\bar{x}$ = sample mean,

• $E$ = margin of error.

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Step 1: Find the Sample Mean ($\bar{x}$)

The mean is the midpoint of the CI:

$$\bar{x}=\frac{\text{Lower Bound}+\text{Upper Bound}}{2}=\frac{3.04+3.61}{2}=\frac{6.65}{2}=3.325$$

Rounded to 2 decimal places:

$$\bar{x}=\mathrm{\$}3.33$$

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Step 2: Find the Margin of Error ($E$)

Subtract the mean from the upper bound (or mean from lower bound):

$$E=\text{Upper Bound}-\bar{x}=3.61-3.325=0.285$$

Rounded to 2 decimal places:

$$E=\mathrm{\$}0.29$$

(Alternatively, $E=\bar{x}-\text{Lower Bound}=3.325-3.04=0.285$)

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Final Answers:

• Sample Mean ($\bar{x}$) = $3.33

• Margin of Error ($E$) = $0.29