One of the advantages that concrete structures have over ste

One of the advantages that concrete structures have over steel structures is fire resistance (which is to say, steel becomes weak when heated). The time, T, that it takes a steel girder to reach a certain temperature (where its strength becomes too low to support its load) in a fire is, however, random. Suppose that for design code development, the following data has been gathered on the time T in hours: 0.736, 0.863, 0.865, 0.913, 0.915, 0.937, 0.983, 1.007, 1.011, 1.064, 1.109, 1.132, 1.140, 1.153, 1.253, 1.394 a)

Compute the values of the sample mean and the sample median.

b) By how much could the smallest sample observation be increased without affecting the value of the sample median?

c) By how much could the smallest sample observation be increased without affecting the value of the sample mean?

d) If you had to express the time T to the developers of the design code using a single number, what number would you use?

I worked out the mean and got 1.03 and the median 0.553

Solution

a) The sample mean is 1.0297 and the sample median is 1.009.

Check the R code for it:

> x <- c( 0.736, 0.863, 0.865, 0.913, 0.915, 0.937, 0.983, 1.007, 1.011, 1.064, 1.109, 1.132, 1.140, 1.153, 1.253, 1.394)
> mean(x)
[1] 1.029687
> median(x)
[1] 1.009

b) The smallest sample observation is 0.736 while the second smallest sample observation is 0.863. Thus, the median value won\'t be affected until the smallest sample observation is increased to be greater than 0.863 or by more than 0.127.

c) Any change in any observation affects the value of sample mean.

d) In the current case the median and mean are close enough. Any single number between the two can be used.