Suppose you are told that a 95% confidence interval for the average price of a gallon of regular gasoline in your state is from $2.85 to $3.84. 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.)
To compute the sample mean (x) and the maximal margin of error (E) from the given confidence interval, we can use the formula:
x - E = Lower limit of the confidence interval
x + E = Upper limit of the confidence interval
Let's denote the lower limit as L and the upper limit as U:
L = $2.85
U = $3.84
From the formula above, we can set up the following two equations:
x - E = L
x + E = U
To solve for x (the sample mean), we can add the two equations together:
(x - E) + (x + E) = L + U
Simplifying the equation:
2x = L + U
2x = $2.85 + $3.84
2x = $6.69
Divide both sides by 2 to solve for x:
x = $6.69 / 2
x = $3.345
Therefore, the sample mean is $3.345.
To find the maximal margin of error (E), we can subtract the lower limit from the sample mean or subtract the sample mean from the upper limit:
E = x - L
E = $3.345 - $2.85
E = $0.495
Therefore, the maximal margin of error is $0.495.
To compute the sample mean and the maximal margin of error (E) from the given confidence interval, we need to find the average price (x) and the difference between the average price and the lower bound of the confidence interval.
Given:
Confidence interval: x − E to x + E
Lower bound of the confidence interval: $2.85
Upper bound of the confidence interval: $3.84
The sample mean (x) is the midpoint of the confidence interval, which is the average of the lower bound and upper bound:
x = (Lower bound + Upper bound) / 2
= ($2.85 + $3.84) / 2
= $3.345
So, the sample mean is $3.345.
The maximal margin of error (E) is half the width of the confidence interval, which is the difference between the sample mean and the lower bound:
E = x - Lower bound
= $3.345 - $2.85
= $0.495
So, the maximal margin of error is $0.495.