91Ó°ÊÓ

The proportion of flights that have an on-time arrival is equal to 80.3%. The sample size is equal to 1000. The confidence level is given to be equal to 99%.

The formula for computing the confidence interval estimate of the population proportion of flights that arrive ontime is given below:

\(CI = \left( {\hat p - E,\hat p + E} \right)\)

Here,\(\hat p\)is the sample proportion, and E is the margin of error.

\(E = {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} \)

The sample proportion of flights that arrive on time is calculated in the following manner:

\(\begin{array}{c}\hat p = 80.3\% \\ = \frac{{80.3}}{{100}}\\ = 0.803\end{array}\)

The sample proportion of flights that do not arrive on time is is calculated in the following manner:

\(\begin{array}{c}\hat q = 1 - \hat p\\ = 1 - 0.803\\ = 0.197\end{array}\)

The sample size is given as n=1000.

The confidence level is given to be equal to 99%. Thus, the level of significance is equal to 0.01.

The corresponding value of \({z_{\frac{\alpha }{2}}}\) is equal to 2.5758.

The margin of error is calculated in the following manner:

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} \\ = 2.5758\sqrt {\frac{{\left( {0.803} \right)\left( {0.197} \right)}}{{1000}}} \\ = 0.0324\end{array}\)

The 99% confidence interval isas follows:

\(\begin{array}{c}CI = \left( {\hat p - E,\hat p + E} \right)\\ = \left( {0.803 - 0.0324,0.803 + 0.0324} \right)\\ = \left( {0.7706,0.8354} \right)\\ = \left( {77.06\% ,83.54\% } \right)\end{array}\)

\( = \left( {77.1\% ,83.5\% } \right)\)

Therefore, the 99% confidence interval estimate of the population percentage of flights that arrive on time is equal to (77.1%,83.5%).