Answer:
(a) (2.573, 3.167) is the 99% confidence interval for the true mean calcium level of the population of people who experienced an unexplained episode of vitamin D intoxication.
(b) There is a 99% probability that the average calcium level of all people who have experienced an unexplained episode of vitamin D intoxication is in the interval.
Step-by-step explanation:
If we have a random sample of size n from a normal distribution with mean and standard deviation
, then we know that
is normally distributed with mean
and standard deviation
. Therefore we can use
as a pivotal quantity.
We have a sample size of n = 12. The sample mean is = 2.87 mmol/l and the standard deviation is
. The confidence interval is given by
where
is the
th quantile of the standard normal distribution. As we want the 99% confidence interval, we have that
and the confidence interval is
where
is the 0.5th quantile of the standard normal distribution, i.e.,
. Then, we have
and the 99% confidence interval is given by (2.573, 3.167)
(b) We found a 99% confidence interval for the true mean calcium level of the population of people who experienced an unexplained episode of vitamin D intoxication. Therefore, there is a 99% probability that the average calcium level of all people who have experienced an unexplained episode of vitamin D intoxication is in the interval.