Which of the following are suitable null hypotheses. If not, explain why.

a. Comparing two groups

Consider comparing the average blood pressure of a group of subjects, both before and after they are placed on a low salt diet. In this case, the null hypothesis is that a low salt diet does reduce blood pressure, i. e., that the average blood pressure of the subjects is the same before and after the change in diet.

b. Classification.

Assume there are two classes, labeled + and -, where we are most interested in the positive class, e. g., the presence of a disease. H0 is the statement that the class of an object is negative, i. e., that the patient does not have the disease.

c. Association Analysis

For frequent patterns, the null hypothesis is that the items are independent and thus, any pattern that we detect is spurious.

d. Clustering

The null hypothesis is that there is cluster structure in the data beyond what might occur at random.

e. Anomaly Detection

Our assumption, H0, is that an object is not anomalous.

Only the null hypothesis stated in the statement d is wrong. The explanation is provided below.

Step-by-step explanation:

In hypothesis testing, especially one comparing two sets of data, the null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test. It usually maintains that, with random chance responsible for the outcome or results of any experimental study/hypothesis testing, its statement is true.

The alternative hypothesis usually confirms the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test. It usually maintains that significant factors other than random chance, affect the outcome or results of the experimental study/hypothesis testing and result in its own statement.

Taking the statements, one at a time

A. Comparing two groups

Consider comparing the average blood pressure of a group of subjects, both before and after they are placed on a low salt diet. In this case, the null hypothesis is that a low salt diet doesn't reduce blood pressure, i.e., that the average blood pressure of the subjects is the same before and after the change in diet.

This null hypothesis is correct. As it is clearly stating that there is no significant difference between the two groups being compared.

b. Classification.

Assume there are two classes, labeled + and -, where we are most interested in the positive class, e.g., the presence of a disease. H0 is the statement that the class of an object is negative, i.e., that the patient does not have the disease.

The study is interested in positive class, hence, this is the theory to be tested that will serve as the alternative hypothesis. The null hypothesis will rightly take on the opposite of that theory or the status quo, which according to random chance is for the class of a patient to be negative. The null hypothesis for the question is correct.

c. Association Analysis

For frequent patterns, the null hypothesis is that the items are independent and thus, any pattern that we detect is spurious.

This Analysis aims to detect patterns in items. So, existence of patterns will serve as the alternative hypothesis and the hypothesis that items are independent with pattern detection not due to a random chance will serve as the null hypothesis. This is correct too.

d. Clustering

The null hypothesis is that there is cluster structure in the data beyond what might occur at random.

This test aims to obtain clustered structure in data. So, definitely, obtaining cluster structure in the data beyond what might occur at random is the alternative hypothesis. When what will happen outside of random factors is considered, then such factors become significant. The null hypothesis stated here is wrong. It does not go against the theory to be tested, then it involves significant, non-random factors.

e. Anomaly Detection

Our assumption, H0, is that an object is not anomalous.

This is a test setup to obviously detect anomalous behaviour. So, an anomalous object is definitely due to significant factors rather than random factors and will serve as the alternative hypothesis.

The null hypothesis will rightly assume that with random factors in full play, an object is not anomalous.

Hope this Helps

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

step-by-step explanation:

left be aligning the decimal points is required to not required see the number of digits to the right