The municipality of a large city has a stock of DVDs. Among the $6\%$ of defective DVDs in the entire stock, $98\%$ are removed. It is also admitted that among the non-defective DVDs, $92\%$ are kept in stock; the others are removed.
One of the city's media libraries wonders whether the number of defective DVDs it possesses is not abnormally high. To do this, it performs tests on a sample of 150 DVDs from its own stock which is large enough for this sample to be treated as successive sampling with replacement. On this sample, 14 defective DVDs are detected.
The asymptotic fluctuation interval at the $95\%$ threshold is given by the formula $$\left[ p - 1{,}96 \frac{\sqrt{p(1-p)}}{\sqrt{n}} ; p + 1{,}96 \frac{\sqrt{p(1+p)}}{\sqrt{n}} \right]$$ where $n$ denotes the sample size and $p$ the proportion of individuals possessing the characteristic studied in this population. The validity conditions are: $n \geqslant 30$, $np \geqslant 5$, $n(1-p) \geqslant 5$.
Can we reject the hypothesis that in this media library, $6\%$ of DVDs are defective?