Problem

Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. The results in the screen display are based on a $95 \%$ confidence level. Write a statement that correctly interprets the confidence interval.
TInterval
\[
\begin{array}{l}
(13.046,22.15) \\
x=17.598 \\
S x=16.01712719 \\
n=50
\end{array}
\]
Choose the correct answer below.
A. The limits of $13.05 \mathrm{Mbps}$ and $22.15 \mathrm{Mbps}$ contain $95 \%$ of all of the data speeds at the airports.
B. We have $95 \%$ confidence that the limits of $13.05 \mathrm{Mbps}$ and $22.15 \mathrm{Mbps}$ contain the true value of the mean of the population of all data speeds at the airports.
C. The limits of $13.05 \mathrm{Mbps}$ and $22.15 \mathrm{Mbps}$ contain the true value of the mean of the population of all data speeds at the airports.
D. We have $95 \%$ confidence that the limits of $13.05 \mathrm{Mbps}$ and $22.15 \mathrm{Mbps}$ contain the sample mean of the data speeds at the airports.

Answer

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Answer

\boxed{\text{B. We have 95% confidence that the limits of 13.05 Mbps and 22.15 Mbps contain the true value of the mean of the population of all data speeds at the airports.}}

Steps

Step 1 :The question is asking for the correct interpretation of the confidence interval. This is a theoretical question and does not require any calculations or Python code to solve.

Step 2 :A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. In this case, the confidence interval is (13.046, 22.15) and the confidence level is 95%.

Step 3 :Option A is incorrect because a confidence interval does not contain a certain percentage of the data points from a sample. It is a range of values that is likely to contain the value of an unknown population parameter.

Step 4 :Option B is correct. A 95% confidence interval means that we are 95% confident that the true population mean lies within the interval (13.046, 22.15).

Step 5 :Option C is incorrect because a confidence interval does not guarantee that it contains the true population mean. It only provides a range of values that is likely to contain the true population mean with a certain level of confidence.

Step 6 :Option D is incorrect because a confidence interval is not about containing the sample mean. The sample mean is a known value calculated from the sample data. The confidence interval is about estimating the unknown population mean.

Step 7 :\boxed{\text{B. We have 95% confidence that the limits of 13.05 Mbps and 22.15 Mbps contain the true value of the mean of the population of all data speeds at the airports.}}

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