S2 4.3–S2 4.5 - Hypotheses and critical regions

S2 4.3–S2 4.5 - Hypotheses and critical regions

An unusual sample can challenge a probability model without proving that it is false. S2 4.3 introduces hypotheses and conclusions, S2 4.4 uses a test statistic and critical region to make a decision, and S2 4.5 distinguishes one-tailed and two-tailed tests. You will connect each decision to the original question and to possible model refinement.

Null and alternative hypotheses

A factory usually has a defect probability of 0.1. A new sample contains more defects than expected. Is this evidence of an increase, or ordinary sample variation? A hypothesis test compares the observation with what the existing model predicts.

Null and alternative hypotheses

The null hypothesis, H0H_0, states the population parameter value used as the working model. The alternative hypothesis, H1H_1, states the proposed departure: greater than, less than or different from that value.

Write hypotheses about the population parameter, rather than about a particular observed count or sample proportion. Define the parameter in context. The direction comes from the investigation's claim, chosen before looking at which direction the data happen to favour.

Worked example 1: a proposed increase

A process has defect probability 0.1. An inspector investigates whether the defect probability has increased. Define the parameter and state the hypotheses.

Plan. Name the population probability, keep its existing value in H0H_0, and translate “increased”.

p=P(a randomly selected item is defective),H0:p=0.1,H1:p>0.1.p=\mathrm P(\text{a randomly selected item is defective}), \qquad H_0:p=0.1,\quad H_1:p>0.1.

Answer. These hypotheses test for a higher defect probability.

Check. H1H_1 concerns the process probability, not whether the observed number exceeds 0.1.

Change only the investigation from “increased” to “changed”. Both directions now matter.

Worked example 2: a proposed change

For the same process, an inspector investigates whether the defect probability has changed. Define the parameter and state the hypotheses.

Plan. Retain the same null value and allow departures on either side.

p=P(a randomly selected item is defective),H0:p=0.1,H1:p≠0.1.p=\mathrm P(\text{a randomly selected item is defective}), \qquad H_0:p=0.1,\quad H_1:p\ne0.1.

Answer. These hypotheses test for any change in defect probability.

Check. Both p=0.05p=0.05 and p=0.15p=0.15 satisfy this alternative; only the second satisfies the previous one.

For a Poisson model the parameter would be its population mean λ\lambda, with its time or area interval defined. For example, H0:λ=8H_0:\lambda=8 must specify whether eight refers to one hour or another observation interval.

Test statistics and critical regions

A test statistic is a statistic selected to measure the departure of interest. For a defect probability it may be the number of defective items in a fixed sample; for a Poisson rate it may be the count in a fixed time.

Critical region and significance level

A critical region is the set of values of the test statistic for which H0H_0 is rejected. Its probability is calculated assuming H0H_0 is true. The stated significance level sets a limit for this rejection probability; the region's actual probability is its actual significance level.

The region is chosen before using the observed value to decide. If the observation lies in the critical region, reject H0H_0; otherwise do not reject it. For a discrete count, whole probability masses cannot usually produce exactly the nominal percentage, so the actual significance may be smaller.

Worked example 3: an observation in the region

To test whether a defect probability has increased, use count XX. Under H0H_0, P(X≥10)=0.041\mathrm P(X\ge10)=0.041 and P(X≥9)=0.086\mathrm P(X\ge9)=0.086. Find the largest upper-tail critical region with probability at most 5%, and decide when x=11x=11.

Plan. Compare neighbouring tail probabilities with 0.05, then compare the observation with the cutoff.

0.041≤0.05<0.086.0.041\le0.05<0.086.

Thus X≥10X\ge10 is allowed, but adding the value 9 would exceed the limit. Since 11≥1011\ge10, reject H0H_0.

Answer. Critical region X≥10X\ge10; actual significance 4.1%; there is significant evidence that the defect probability has increased.

Check. The rejected observation is in the upper tail, matching the proposed increase.

Change only the observed count from 11 to 9. The critical region must stay the same.

Worked example 4: an observation outside the region

For the same test and supplied probabilities, decide when x=9x=9.

Plan. Use the previously chosen region, rather than moving it to include the new observation.

9<10,9<10,

so 9 is not in the critical region.

Answer. Do not reject H0H_0. There is insufficient evidence at the 5% level that the defect probability has increased.

Check. The inclusive tail at the observation has probability 0.086, which exceeds 0.05 and agrees with non-rejection.

A critical region is a set such as X≥10X\ge10; P(X≥10)=0.041\mathrm P(X\ge10)=0.041 is a probability statement describing that set. Give the region when asked for the region.

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