Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let denote the average alcohol content
for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.8, 9.4).

a. Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning.
b. Consider the following statement: There is a 95% chance that mu is between 7.8 and 9.4. Is this statement correct? Why or why not?
c. Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.4. Is this statement correct? Why or why not?
d. Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include mu. Is this statement correct? Why or why not?

(a) A 90% confidence interval will be narrower.

(b) The statement "There is a 95% chance that μ is between 7.8 and 9.4" is true.

(c) The statement "We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.4" is correct.

(d) The statement is correct.

Step-by-step explanation:

The (1 - α)% confidence interval for population mean is:

The confidence interval is affected by:

The 95% confidence interval for the population mean alcohol content of all bottles of the brand of cough syrup is (7.8, 9.4).

A 95% confidence interval implies that there a 0.95 probability that the true value of the parameter will be contained in the interval.

Or, there is a 95% confidence that the true value of the parameter will be contained in the interval.

Or, if 100 such samples are selected to construct 100 such 95% confidence interval then 95 of those 100 confidence interval will contain the true parameter value.

(a)

On changing the confidence level the critical value changes which thus changes the confidence interval width.

If the confidence level is decreased to 90% from 95% then the critical value will also decrease. On decreasing the critical value the interval width decreases. Thus, narrowing the confidence interval.

Hence, a 90% confidence interval will be narrower.

(b)

The statement "There is a 95% chance that μ is between 7.8 and 9.4" is true.

Because a 95% confidence interval implies that there is 95% probability or chance that the true value of mean will lie in the interval (7.8, 9.4).

(c)

The statement "We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.4" is correct.

The 95% confidence interval for mean alcohol content implies that there is a 95% confidence that the true value of mean will be contained in the interval (7.8, 9.4).

And a 95% confidence is quite high.

(d)c

Because if 100 sample of size 50 are selected from the same population and the 95% confidence interval is constructed for all these 100 sample then at least 95 of these 100 confidence interval will include the true value of mean alcohol content.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!