Question

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. They are also interested in the variation of the number of babies. Suppose that an airline executive believes the average number of babies on flights is six with a variance of nine at most. The airline conducts a survey. The results of the $18$ flights surveyed give a sample average of $6.4$ with a sample standard deviation of $3.9$. Conduct a hypothesis test of the airline executive’s belief.

Short answer

If we take $\alpha =0.05$, we can see that $p<\alpha$. This means that we reject null hypothesis.

There is sufficient evidence to conclude that the standard deviation is greater than $3$.

Step-by-step solution

Given Information

We want to test these hypotheses:

$\begin{array}{ll}{H}_0:& \sigma \le 3\\ H_1:& \sigma >3\end{array}$

There are 18 flights surveys so $n=18$. We know that the number of degrees of freedom is$n-1=17.$

Explanation

We are using ${\chi }^2$distribution with $17$degrees of freedom to test our hypothesis. We are given $s=3.9$, thus the value of test statistic is:

${\chi }^2=\frac{(n-1)s^2}{{\sigma }^2}=28.73$

Using the applet for ${\chi }^2$distribution, we can find the $p$-value and the result is $0.0371$:

Chi-Square Distribution

$X~{\chi }_{\left(df\right)}^2$