5. Gareth has a keen interest in pop music. He recently read the following claim in a music magazine.
\section*{In the pop industry most songs on the radio are not longer than three minutes.}
- He decided to investigate this claim by recording the lengths of the top 50 singles in the UK Official Singles Chart for the week beginning 17 June 2016. (A 'single' in this context is one digital audio track.)
Comment on the suitability of this sample to investigate the magazine's claim.
- Gareth recorded the data in the table below.
| Length of singles for top 50 UK Official Chart singles, 17 June 2016 |
| 2.5-(3.0) | 3.0-(3.5) | 3.5-(4.0) | 4.0-(4.5) | 4.5-(5.0) | 5.0-(5.5) | 5.5-(6.0) | 6.0-(6.5) | 6.5-(7.0) | 7.0-(7.5) |
| 3 | 17 | 22 | 7 | 0 | 0 | 0 | 0 | 0 | 1 |
\includegraphics[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-4_860_1435_1343_379}
State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table. - Gareth also produced a box plot of the lengths of singles.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Length of single for top 50 UK Official Singles Chart 17 June 2016}
\includegraphics[alt={},max width=\textwidth]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-5_504_812_406_644}
\end{figure}
He sees that there is one obvious outlier.
- What will happen to the mean if the outlier is removed?
- What will happen to the standard deviation if the outlier is removed?
- Gareth decided to remove the outlier. He then produced a table of summary statistics.
- Use the appropriate statistics from the table to show, by calculation, that the maximum value for the length of a single is not an outlier.
| Summary statistics Length of single for top 50 UK Official Singles Chart (minutes) |
| \multirow{2}{*}{Length of single} | N | Mean | Standard deviation | Minimum | Lower quartile | Median | Upper quartile | Maximum |
| 49 | 3.57 | 0.393 | 2.77 | 3.26 | 3.60 | 3.89 | 4.38 |
- State, with a reason, whether these statistics support the magazine's claim.
- Gareth also calculated summary statistics for the lengths of 30 singles selected at random from his personal collection.
| Summary statistics Length of single for Gareth's random sample of 30 singles (minutes) |
| \multirow{2}{*}{Length of single} | N | Mean | Standard deviation | Minimum | Lower quartile | Median | Upper quartile | Maximum |
| 30 | 3.13 | 0.364 | 2.58 | 2.73 | 2.92 | 3.22 | 3.95 |