## Related questions with answers

As worldwide air traffic volume has grown over the years, the problem of airplanes striking birds and other flying wildlife has increased dramatically. The international Journal for Traffic and Transport Engineering (Vol. 3, 2013) reported on a study of aircraft bird strikes at Aminu Kano international Airport in Nigeria. During the survey period, a sample of 44 aircraft bird strikes were analyzed. The researchers found that 36 of the 44 bird strikes at the airport occurred above 100 feet. Suppose an airport air traffic controller estimates that less than 70% of aircraft bird strikes occur above 100 feet. Comn1ent on the accuracy of this estimate. Use a 95% confidence interval to support your inference.

Solution

VerifiedGiven:

$\begin{align*} n&=\text{Sample size}=44 \\ x&=\text{Number of successes}=36 \\ c&=\text{Correlation coefficient}=95\%=0.95 \end{align*}$

Since there are 36 successes in the sample of size 44, there are $44-36=8$ failures in the sample.

Since the number of failures is not at least 15, the large sample conditions is not satisfied and thus we need to use the $\textit{adjusted}$ confidence interval for a population proportion.

The sample proportion is the number of successes divided by the sample size. Since we use the adjusted confidence interval, we add 2 failures and 2 successes to the sample.

$\tilde{p}=\dfrac{x+2}{n+4}=\dfrac{36+2}{44+4}=\frac{38}{48}\approx 0.7917$

For confidence level $1-\alpha=0.95$, determine $z_{\alpha/2}=z_{0.025}$ using the normal probability table in the appendix, which is the z-score corresponding to a probability of $0.5-\alpha/2=0.475$:

$z_{\alpha/2}=1.96$

The margin of error is then:

$E=z_{\alpha/2}\cdot \sqrt{\dfrac{\tilde{p}(1-\tilde{p})}{n+4}}=1.96\times \sqrt{\dfrac{0.7917(1-0.7917)}{48}}\approx 0.1149$

The boundaries of the confidence interval are then:

$\tilde{p}-E=0.7917-0.1149=0.6768$

$\tilde{p}+E=0.7917+0.1149=0.9066$

We are 95% confident that the true proportion of bird strikes at the airport that occurred above 100 feet is between 0.6768 and 0.9066.

Since the confidence interval $(0.6768, 0.9066)$ contains values above $0.7=70\%$, it is likely that the true proportion of bird strikes at the airport that occurred above 100 feet is larger than 0.7 and thus the estimate that less than 70% of aircraft bird strikes occur above 100 feet appears to be inaccurate.

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