Question

An experiment to study the lifetime (in hours) for acertain type of component involved putting ten components into operation and observing them for 100hours. Eight of the components failed during that

period, and those lifetimes were recorded. Denote the lifetimes of the two components still functioning after100 hours by 100+. The resulting sample observations were

48 79 100+ 35 92 86 57 100+ 17 29

Which of the measures of center discussed in this section can be calculated, and what are the values of those measures? (Note:The data from this experiment is said to be “censored on the right.”)

Short answer

The median value is 68.

The 20% trimmed mean is 66.2.

The 30% trimmed mean is 67.5.

Step-by-step solution

Given information

The sample data for the lifetime (in hours) for a certain type of component is provided.

Computing the measures of center

The measures of the center that can be computed in the provided scenario are median, 20% trimmed mean and 30% trimmed mean.

The sample median is computed by first ordering the data in ascending order.

The data arranged in ascending order is as follows,

17 29 35 48 57 79 86 92 100+ 100+

For the even number of observations, the median value is computed as,

\(\begin{array}{c}\tilde x &=& average\;of\;{\left( {\frac{n}{2}} \right)^{th}}and\;{\left( {\frac{n}{2} + 1} \right)^{th}}\;ordered\;values\\ &=& average\;of\;{\left( {\frac{{10}}{2}} \right)^{th}}and\;{\left( {\frac{{10}}{2} + 1} \right)^{th}}\;ordered\;values\\ &=& average\;of\;{\left( 5 \right)^{th}}and\;{\left( 6 \right)^{th}}\;ordered\;values\\ &=& \frac{{57 + 79}}{2}\\ &=& 68\end{array}\)

Thus, the median value is 68.

Using the arranged data,

The 20% trimmed mean is computed by eliminating the smallest two and the largest two values from the data and taking the average of the rest of the data.

The 20% trimmed mean is given as,

\(\begin{array}{c}{{\bar x}_{tr\left( {20} \right)}} = \frac{{35 + 48 + 57 + ... + 92}}{6}\\ = 66.2\end{array}\)

Therefore, the 20% trimmed mean is 66.2.

The 30% trimmed mean is computed by eliminating the smallest three and the largest three values from the data and taking the average of the rest of the data.

\(\begin{array}{c}{{\bar x}_{tr\left( {30} \right)}} &=& \frac{{48 + 57 + 79 + 86}}{4}\\ &=& 67.5\end{array}\)

Therefore, the 30% trimmed mean is 67.5.