01 · Count the event
List the counts included in the evidence probability for observed x=14 and an upper-tailed test.
Hint
Keep the observed count.
Worked solution
14,15,16,17,18,19,20.
Understand · explore · practise
Significance and tail evidence
Understand why a hypothesis test uses an observed-or-more-extreme tail, interpret a significance threshold and avoid confusing evidence with the probability a claim is true.
Before you startBinomial tails, hypotheses and choosing a test direction.
01 / Ask how unusual the evidence is under H₀
The calculation assumes the baseline; it does not calculate its truth.
For a one-sided test, include observed and more extreme counts.
An upper alternative uses P(X≥x); a lower alternative uses P(X≤x), calculated under H₀.
Here X is the random count and x is the observed count. For the stated one-sided test, this tail probability is the p-value. A small value means such results would be unusual under the null model.
One bar or the whole evidence tail?Explore
Under H₀, X~B(20,0.5). The preselected alternative is p>0.5. Choose the observed count.
02 / A single outcome can be rare without its tail being rare
Do not substitute the point probability P(X=x).
Watch: one count compared with the full upper tail
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Under H₀, X~B(20,0.5). With H₁: p>0.5 and x=14, test at 5%.Worked example
P(X=14)≈0.03696442
This is one bar, not the whole evidence tail.
P(X≥14)≈0.05765915
Include counts 14,15,…,20.
0.05765915>0.05: do not reject H₀
There is insufficient evidence at 5% that p exceeds 0.5. The point probability would give the wrong decision.
02 · Correct the calculator choice
A student uses binomial point probability at 14. Which calculation should replace it?
Hint
Use a cumulative complement with the correct boundary.
Worked solution
P(X≥14)=1−P(X≤13). A cumulative value through 14 would exclude the observed count from the complement.
03 / Compare with the stated significance level
Use a consistent decision rule and retain precision.
In these lessons, reject H₀ when the appropriate one-sided tail probability is at most α. For α=0.05, this means ≤0.05. If a question explicitly prescribes a different strict convention, follow it. Keep enough digits to decide the comparison before rounding the displayed answer.
03 · Stronger observation
For the same upper-tailed test, x=15 gives P(X≥15)≈0.02069473. State the decision at 5%.
Hint
Compare with 0.05.
Worked solution
Reject H₀. There is sufficient evidence at 5% that the success probability is greater than 0.5, under the model assumptions.
04 · Change the level
Would that x=15 result reject at 1%?
Hint
Use 0.01 instead.
Worked solution
No: 0.02069473>0.01. There is insufficient evidence at 1%. The same observation can lead to different decisions at different prespecified levels.
05 · Exact equality
Under the at-most convention, what happens if a one-sided p-value is exactly0.05 at a 5% level?
Hint
Equality belongs to≤.
Worked solution
Reject H₀. An approximately displayed 0.05 is not enough to establish exact equality; retain calculation precision.
04 / Read the probability in the right direction
Unusual under a model is different from probability of that model.
A tail probability is calculated conditional on H₀ and the sampling assumptions. It is not P(H₀ is true), nor the probability that a particular conclusion is wrong. A5% significance rule describes the rejection behaviour of the procedure under the null; the actual binomial rejection probability may be smaller because counts are discrete.
06 · Misread p-value
A p-value is 0.02. Does this mean a 2% probability that H₀ is true?
Hint
Which assumption was used to obtain the probability?
Worked solution
No. It means an observed-or-more-extreme result has probability 0.02 under the specified null model for this test. It does not assign a probability to H₀.
07 · Misread significance
Does a 5% level mean a 95% probability that the alternative is true after rejection?
Hint
The level belongs to a procedure, not a posterior probability.
Worked solution
No. It controls the test’s false-rejection probability under H₀, subject to its construction and assumptions. It is not a probability that the alternative is true.
05 / Use the lower tail for a decrease
A complement is not always needed.
Under H₀, X~B(20,0.5), but the preselected alternative is p<0.5. Observe x=5.Worked example
Use P(X≤5), not P(X≥5)
Small counts point in the alternative direction.
Symmetry gives P(X≤5)=P(X≥15)≈0.02069473
This equals the corresponding upper-tail probability for the fair binomial model.
At5%, reject H₀
There is sufficient evidence that p is below 0.5.
08 · Wrong tail
For a lower-tailed test with observed x=6, which probability is needed?
Hint
Include six and all smaller counts.
Worked solution
P(X≤6). In this fair 20-trial model it equals P(X≥14)≈0.05765915, so it would not reject at 5%.
09 · Opposite direction
If the preselected alternative is p>0.5 but x=5 is observed, can its small lower-tail probability justify rejection for an increase?
Hint
The claim points upwards.
Worked solution
No. An upper-tailed test uses P(X≥5), which is large. A low result does not provide evidence for an increase.
06 / Keep two-tailed allocation separate
A full two-tailed test needs a convention for both ends.
For the equal-tail method used later, compare the probability in the observed tail with α/2. Thus a 5% two-tailed test compares with 0.025 at the relevant end. This lesson’s interactive decisions are explicitly one-sided at 5%; do not reuse them unchanged for a two-tailed test.
10 · Half the threshold
At a 5% equal-tail test, an observed-tail probability is 0.03. Reject or not?
Hint
Compare with 0.025.
Worked solution
Do not reject:0.03>0.025, even though 0.03 is less than 0.05.
11 · Another observation
At that same two-tailed level, an observed-tail probability is 0.02. What is the decision?
Hint
Use the half-level.
Worked solution
Reject H₀ under the stated equal-tail method, since 0.02≤0.025.
07 / Give evidence its proper limits
A decision is not a proof or a measure of practical importance.
12 · Nonrejection
Complete the sentence: “We do not reject H₀, therefore…”
Hint
Avoid claiming equality has been established.
Worked solution
…there is insufficient evidence against H₀ at the stated level under the model. It does not prove the baseline probability is correct.
13 · Practical importance
Does a small p-value tell us that an effect is large or important in practice?
Hint
A tail probability is not an effect-size measurement.
Worked solution
No. Describe the observed change and context separately; statistical evidence and practical importance are different questions.
14 · Model conditions
Why can a precisely calculated tail still be misleading for strongly dependent observations?
Hint
What distribution justified the calculation?
Worked solution
The binomial distribution assumes independence and a common probability. If those assumptions fail, its tail may not describe the sampling process, even when the arithmetic is correct.
08 / Model, event, probability, comparison, conclusion
Every decision needs the complete chain.
Calculate under H₀. Select the tail from the prechosen alternative and include the observed count. Compare with the appropriate stated level using sufficient precision. Conclude in context without assigning a probability to the hypothesis or claiming proof.
Section 1 of 8 · Ask how unusual the evidence is under H₀