Question

A research study is comparing the aorta diameters of patients who have aortic disorders to the aortas of those who do not have any disorders. The aorta diameters of 15 cases (those with aortic disorders) were measured. The results are given below, measured in millimeters (mm):

7.22, 9.24, 12.01, 7.18, 9.17, 8.26, 14.03, 10.55, 11.09, 10.17, 8.56, 8.29, 7.87, 11.24, 13.14

a) Because the sample size is small, to compute a confidence interval for the mean, we must assume that the population distribution of the aorta diameters for cases is approximately normally distributed. Describe in words what you would do to decide whether or not this assumption is reasonable. There is no need to actually check if the data is normally distributed, just describe what you would do to determine this.

b) Let us assume that all necessary assumptions are met, find and clearly state in a sentence a 99% confidence interval for the true population mean aorta for all individuals with aortic disorders.

c) Suppose we know that the population mean aorta diameter for individuals without aortic disorders is μ = 8.17 mm. Does the confidence interval obtained in (b) suggest that patients with aortic disorders tend (on average) to have different aorta diameters than those without aortic disorders? Briefly explain your answer.

Answer 1

(a)

We have to calculate mean () and standard deviation () of the distribution using sample data. Then we shall compute probabilities in 1-, 2- and 3- intervals about mean (). If the probabilities are approximately same as that in case of standard normal variate i.e. 0.6826895, 0.9544997 and 0.9973002 respectively, then we can conclude that the assumption of normal distribution is satisfactory.

(b)

We assume that necessary assumptions to be normally distributed are met.

Suppose, random number X denotes the aorta diameter.

Mean is given by

Standard deviation is given by

We know,

So, 99% confidence interval is given by

i.e.

(c)

We are given with the data that mean aorta diameter for individuals without aortic disorder is which lies in our 99% confidence interval.

We could conclude that mean aorta diameter for individuals without aortic disorder being different from that of individuals with aortic disorder at 99% confidence interval if the said value i.e. lied beyond the interval . But this does not happen here.

Hence, based on our calculation from the observed data we conclude that patients with aortic disorder does not tend to have different arota diameters than those without aortic disorders.