Question

You have collected the following data on internal pressure strength of 16 glass bottles. (a) Does it seem reasonable to conclude that the internal pressure strength is normally distributed? 226.16208.15 211.14 221.31 02 ,201 .6 2M 219.54193.71 188.12 202.21 193.73 200.81 224.39 201.63 195.45 203

Answer 1

a) The number of samples we have is an important criteria is deciding if this is normally distributed. It is recommended to have 30 samples or more for this. Here, we have only 16 samples, so any conclusion drawn from this will be premature.

Analysing further, we see that the sample mean $\mu_s = \frac{\sum x_i}{n} = \frac{3296.72}{16} \approx 206.05$ and $\sigma_s = \sqrt{\frac{\sum (x_i - \mu_s)^2}{n - 1}} = \sqrt{\frac{2008.55}{15}} \approx 11.57$

We now check how many points lie within 1, 2, and 3 standard deviations of the sample mean.

$206.05 \pm 11.57 = [194.48, 217.62]$ - 9/16 = 56.3%

$206.05 \pm 2*11.57 = [182.91, 229.19]$ - 16/16 = 100%

So, assuming that sample mean is the actual mean, we see that 56.3% of the samples lie within 1 standard deviation of mean and 100% of the samples lie within 2 standard deviations of mean. For a normal distribution, we expect 68.3% of the samples to lie within 1 standard deviation of the mean and 95.4% of the samples to lie within 2 standard deviations of the mean. The difference between the two seems to be significant.

So, the small sample size and the significant difference in the distribution between the sample and the expected normal distribution implies that we cannot conclude that the internal pressure strength is normally distributed.