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Question Find the margin of error and sample mean using the given 95%95\% confidence interval (1.58,2.06)(1.58, 2.06). The margin of error is $0.24$.Thesamplemeanis\$0.24\$. The sample mean is \$1.82\$.

Studdy Solution

STEP 1

Assumptions
1. The confidence interval is (1.58, 2.06)
2. The confidence interval is symmetric around the sample mean

STEP 2

The confidence interval is defined as the range in which we are confident that the population mean lies. It is symmetric around the sample mean. So, the sample mean is the midpoint of the confidence interval.
Sample mean=(Lower limit+Upper limit)/2Sample\, mean = (Lower\, limit + Upper\, limit) / 2

STEP 3

Now, plug in the given values for the lower limit and upper limit to calculate the sample mean.
Sample mean=(1.58+2.06)/2Sample\, mean = (1.58 + 2.06) / 2

STEP 4

Calculate the sample mean.
Sample mean=(1.58+2.06)/2=1.82Sample\, mean = (1.58 + 2.06) / 2 = 1.82

STEP 5

The margin of error is defined as the distance from the sample mean to either end of the confidence interval. So, we subtract the sample mean from the upper limit (or subtract the lower limit from the sample mean, the result will be the same due to the symmetry of the confidence interval).
Margin of error=Upper limit−Sample meanMargin\, of\, error = Upper\, limit - Sample\, mean

STEP 6

Now, plug in the given values for the upper limit and the calculated sample mean to calculate the margin of error.
Margin of error=2.06−1.82Margin\, of\, error = 2.06 - 1.82

STEP 7

Calculate the margin of error.
Margin of error=2.06−1.82=0.24Margin\, of\, error = 2.06 - 1.82 = 0.24
The margin of error is 0.24. The sample mean is 1.82.

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