Question

8.1.3 A random sample has been taken from a normal distribution and the following confidence intervals constructed using the same data: (38.02, 61.98) and (39.95, 60.05)
(a) What is the value of the sample mean?
(b) One of these intervals is a 95% CI and the other is a 90% CI. Which one is the 95% CI and why?

Answer 1

Answer:

a) The value of the sample mean is 50

b) (38.02, 61.98) is the 95% as it covers more values than the other.

Step-by-step explanation:

a) Upper and lower values for the confidence intervals are at the same distance from the mean, as they represent an upper and lower limit for the value, and the most probable value is the one in the middle, which, in a normal distribution is the mean.

As they derive from the same data, both pairs of values should give us the same mean:

$\mbox {mean}=\frac{\mbox{upper value+lower value}}{\mbox{2} }\\\mbox {mean from the first set}=\frac{38.02+61.98}{2}=50 \\\mbox {mean from the first set}=\frac{39.95+60.05}{2}=50$

b) The percentage of the confidence interval tells us which is the area below the bell curve (of the normal distribution) between the limits given.

One could give a value with a 100% CI, but limits should be minus and plus infinity, and as the interval gets smaller, less values are covered (less area), and less certain we are about our value being inside the interval.

Then, we can say that the widest interval must be the 95% CI, i.e. (38.02, 61.98).