Answer:
The 98% confidence interval for the sample mean is (66.763, 76.969).
Step-by-step explanation:
The main difficulty of this problem is to write all the data into an Excel sheet. Once we have done this part the problem is not difficult. Here we are assuming that we do not know the standard deviation. Also, we need to remark that the data has been written from the cell A1 to the cell A26.
<em>First step</em>: Let us calculate the mean of the sample. This can be done using an Excel function: AVERAGE. If we want to write the mean in the cell B1, we mark the cell and then write =AVERAGE(A1:A26). Notice that the arguments of the function is the first and the last cell of our data. With this we get that μ = 71.86538462
, and rounding μ = 71.866.
<em>Second step</em>: Now we need to calculate the standard deviation of the sample, because we do not know the theoretical standard deviation. This can be done using an Excel function: STDEV. If we want to write the mean in the cell B2, we mark the cell and then write =STDEV(A1:A26). Notice that the arguments of the function is the first and the last cell of our data. With this we get that σ = 11.0160226 and rounding σ = 11.016.
<em>Third step: </em>We are going to calculate the confidence. This can be done using an Excel function: CONFIDENCE. If we want to write the mean in the cell B3, we mark the cell and then write =CONFIDENCE(0.02;11.016;26). Let us explain what are the arguments of the function CONFIDENCE:
- The number 0.02 is the level of confidence. Notice that in the statement of the problem we were asked to find ‘‘the 98% confidence interval’’, but Excel can not understand this data, so we need to ‘‘normalize’’ it using the formula 1 - 98/100 = 1-0.98=0.02.
- The second number, 11.016, is the standard deviation obtained in the second step.
- The 26 is the number of samples we have.
With this we get that the confidence is ε=5,10298125
, and rounding is ε=5,103.
<em>Fourth step</em>: Finally we are going to find the confidence interval. Here we are going to use the results of the first and third step. The confidence interval is obtained by the formula (μ - ε, μ + ε). Then,
(μ - ε, μ + ε) = (71.866 - 5,103, 71.866 + 5,103) = (66.763, 76.969)
is the first term of the series,
is the common ratio of the series, and
is the number of terms in this series.