Hypotheses

Hypotheses Described

Hypotheses are statements about the world and how things work. They express a relationship between a set of measurements and either a second set of measurements or a standard value (such as a product specification). They should be simple, specific and testable, and they should be stated before you start testing. A good, short overview can be found in Choudhury et al.

Hypotheses consist of a null hypothesis—what we assume is true, and then try to disprove with data—and an alternate hypothesis—the opposite of the null hypothesis, which must be true if the null hypothesis is false. The null hypothesis always assumes that there is no detectable difference between the things being measured. The alternate hypothesis is that there is a difference. These relationships can be equals versus not equals, or they can be less-than-or-equal-to versus greater than, or they can be greater-than-or-equal-to versus less than.

There are three basic types of hypothesis: hypotheses about the averages (or means) of a value; those about the variation (or standard deviation) in a value and those about proportions. In the case of means and proportions, the null hypothesis is expressed as the difference between two values. In the case of standard deviations, the null hypothesis is expressed as the ratio between two values.

We don’t have to assume differences of zero (that two values are equal); we can select a difference that we want to detect. Likewise, we don’t have to assume a ratio of one for proportions (that the two proportions are equal); we can use other values for the ratio of proportions that we’re interested in.

Relationships

Table 1: Summary Relationships
Measure to Compare If the Null Hypothesis is… Then the Alternative Hypotheses is…
Means μ μ(A) – μ(B) = δ
for any difference δ
μ(A) – μ(B) ≠ δ
μ(A) – μ(B) ≤ δ μ(A) – μ(B) > δ
μ(A) – μ(B) ≥ δ μ(A) – μ(B) < δ
Standard Deviations σ
for any difference r
Proportions p p(A) – p(B) = δ
for any difference δ
p(A) – p(B) ≠ δ
p(A) – p(B) ≤ δ p(A) – p(B) > δ
p(A) – p(B) ≥ δ p(A) – p(B) < δ

Example Hypotheses

Table 2: Sample hypotheses
Comparison Null Hypothesis (H0) Alternative Hypothesis (HA)
Means, two sided

The average torque that will break a bolt from supplier A is the same as that needed for the bolt from supplier B.

μ(A) – μ(B) = 0 Nm

the average torque that will break a bolt from supplier A is different than that needed for the bolt from supplier B.

μ(A) – average torque μ(B) ≠ 0 Nm

Mean vs. a specification, one-sided

The average torque to break a bolt must be greater than 100 Nm (74 ft-lbs).

μ(A) – 100 Nm ≥ 0 Nm

The average torque to break a bolt is less than 100 Nm.

μ(A) – 100 Nm < 0

Standard deviation, one-sided

The standard deviation in torque needed to break a bolt from supplier A is up to 2 Nm greater than the standard deviation in torque for the bolt from supplier B.

The standard deviation in torque needed to break a bolt from supplier A is greater than 2 Nm more than the standard deviation in torque for the bolt from supplier B.

Proportions, one-sided

The scrap rate of the new process (A) is not improved compared to the old process (B).

p(A) – p(B) ≥ 0.5

The scrap rate of the new process is less than half the scrap rate of the old process.

p(A) – p(B) < 0.5

The equations above are courtesy of the excellent online LaTeX editor at Codecogs.

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