Question 698306
The t-statistic for this data is {{{t = (x - mu)/(s/sqrt(n))}}}, where x is the mean of the sample of bulbs, mu is the population mean, s is the standard deviation of the sample, and n is the number of items in the sample. Here, x = 290, mu = 300, s = 50, and n = 15, so:<br>

{{{t = (290 - 300)/(50/sqrt(15)) = -.7746}}}. So, the t-score is -.7746.<br>

The t-distribution changes in shape depending on how many items are in the sample. You have to use the t-distribution corresponding to the appropriate number of degrees of freedom. For probability calculations, the number of degrees of freedom is n - 1, so here you need the t-distribution with 14 degrees of freedom.<br>

You'll need either a table or a utility to calculate the probability that t < -.7746 with 14 degrees of freedom. I use http://stattrek.com/online-calculator/t-distribution.aspx as a calculator for this. This calculator returns the probability that, assuming the population mean is true, the t-value is less than the t-value obtained With 14 degrees of freedom and a t score of -.7746, the table returns a probability of the bulbs lasting less than 290 days on average of .2257 assuming the mean life of the bulbs is 300 days.