Answer:
The point estimate that should be used in constructing the confidence interval is 3.5.
The 95% confidence interval for the true mean difference between the mean height of the American students and the mean height of the non-American students, in inches, is (2.25, 4.75).
Step-by-step explanation:
Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
American students:
Sample of 12, mean height of 68.4 inches with a standard deviation of 1.64 inches. This means that:


Non-American students:
Sample of 17, mean height of 64.9 inches with a standard deviation of 1.75 inches. This means that:


Distribution of the difference:


The point estimate that should be used in constructing the confidence interval is 3.5.
Confidence interval:

In which
z is the z-score that has a p-value of
.
95% confidence level
So
, z is the value of Z that has a p-value of
, so
.
The lower bound of the interval is:

The upper bound of the interval is:

The 95% confidence interval for the true mean difference between the mean height of the American students and the mean height of the non-American students, in inches, is (2.25, 4.75).