Hypothesis tests 201-SN1-RE: revision sheet

Q: How do you know whether a hypothesis test is one-tailed or two-tailed?

You decide from the question asked, before looking at the data. If the study wants to establish a specific increase or decrease, with words such as increases, exceeds, reduces or is lower, the test is one-tailed on the side of that word. If it only asks whether the value has changed or differs, in either direction, it is two-tailed. With no indication of direction, take the two-tailed test.

Q: When should you use Student's t distribution rather than the normal distribution in a test on a mean?

When the sample has fewer than 30 observations and the population standard deviation is unknown, you use Student's t distribution with n minus 1 degrees of freedom, provided the population is assumed normal. With 30 observations or more, the central limit theorem allows the normal distribution, even if the data are not normal. Student's critical value is always larger than that of the normal distribution.

Q: What does a p-value of 0.03 mean?

It means that, if the null hypothesis were true, a result at least as far from that hypothesis as the one observed would occur in only 3 percent of samples. So the null hypothesis is rejected at the 5 percent level, but not at the 1 percent level. It is not the probability that the null hypothesis is true.

Q: Can you accept the null hypothesis when the test does not reject it?

No. Not rejecting the null hypothesis only means that the data are not far enough from it to rule out chance. A real but small effect, or a sample that is too small, often gives this result. So you write that the data do not allow you to conclude, at the chosen level, what the study wanted to establish.

The thread of the chapter

Everything is computed under $H_{0}$ , and nothing proves $H_{0}$ : the statistic measures, in standard errors, the distance from the data to the null hypothesis, and only a distance larger than the level set in advance allows a conclusion.

Topics covered

Null and alternative hypotheses
One-tailed and two-tailed tests
Significance level
Critical region
Test statistic
P-value
Student's t distribution
Test on a mean
Test on a proportion
Type I error
Type II error
Power of a test

The essentials

The logic of a test: reasoning under the null hypothesis

• $H_{0}$ carries the equality on a PARAMETER: $\mu=\mu_{0}$ or $p=p_{0}$ . $H_{1}$ carries what the study wants to establish: $\neq$ (two-tailed), $<$ or $>$ (one-tailed).
• We ASSUME $H_{0}$ is true and compute the distribution of the statistic under this hypothesis: centred on $\mu_{0}$ , with the standard error of $H_{0}$ .
• BEFORE the data, we set the level $\alpha$ , hence the critical region: the values of the statistic that would be too rare if $H_{0}$ were true.
• Statistic in the critical region: we reject $H_{0}$ . Otherwise, we do not reject it, which does not mean it is true.

Under

H_{0}

Z

exceeds

1.645

5\%

of cases:

z=2.30

falls in this orange zone and leads to rejecting

H_{0}

z=1.20

does not.

A test reasons like a proof by contradiction, but a statistical one: it shows that the data are unlikely under H₀, never that H₀ is true.

The three statistics of the course, and their conditions

• Mean, $n\geq 30$ : $z=\dfrac{\overline{x}-\mu_{0}}{s/\sqrt{n}}$ , normal distribution by the central limit theorem, with $s$ replacing $\sigma$ .
• Mean, $n<30$ , population assumed NORMAL: $t=\dfrac{\overline{x}-\mu_{0}}{s/\sqrt{n}}$ , Student's t distribution with $n-1$ degrees of freedom.
• Proportion, $np_{0}\geq 5$ and $n(1-p_{0})\geq 5$ : $z=\dfrac{\hat{p}-p_{0}}{\sqrt{p_{0}(1-p_{0})/n}}$ , with $p_{0}$ and not $\hat{p}$ .
• In all three cases, a randomly drawn sample and independent observations (sampling with replacement, or a population large compared with $n$ ).

The conditions are written on the paper before the calculation: a forgotten condition costs the mark even when the statistic is right.

The two errors, and what controls them

• Type I error: rejecting $H_{0}$ when it is true. Its probability is $\alpha$ , set by the experimenter.
• Type II error: not rejecting $H_{0}$ when it is false. Its probability $\beta$ depends on the true value of the parameter.
• Power $=1-\beta$ : the probability of detecting a gap that exists. For a fixed $n$ , lowering $\alpha$ increases $\beta$ ; increasing $n$ reduces $\beta$ with $\alpha$ fixed.

The rules in table form

Each row reads from left to right: the assumptions, then the result. A red cell is not an answer, it is the finding that the form settles nothing and the instruction to rewrite it. Every case is followed by a worked example.

When to reject $H_{0}$ at the $5\%$ level

Read the distribution (normal or Student) and the form of $H_{1}$ ; the last column gives the rejection rule. The two red rows are rules that do not exist.

Distribution	$H_{1}$	Reject $H_{0}$ if
normal	$\mu\neq\mu_{0}$	$\|z\|>1.96$
Example: $z=-2.05$ : $\|z\|>1.96$ , we reject.
normal	$\mu>\mu_{0}$	$z>1.645$
Example: $z=1.80>1.645$ : we reject, even though $1.80<1.96$ .
normal	$\mu<\mu_{0}$	$z<-1.645$
Example: $z=-1.50>-1.645$ : we do not reject.
Student, $9$ df	$\mu\neq\mu_{0}$	$\|t\|>2.262$
Example: $n=10$ , $t=2.10<2.262$ : we do not reject, even though $2.10>1.96$ .
Student, $9$ df	$\mu>\mu_{0}$	$t>1.833$
Example: $n=10$ , $t=1.90>1.833$ : we reject.
normal	$\mu>\mu_{0}$	$\|z\|>1.96$ no such rule
Example: $z=1.80$ : this rule does not reject, the correct rule does. What to do: One-tailed: the whole area $\alpha$ is in ONE tail, and the critical value is $1.645$ at $5\%$ .
Student, $9$ df	$\mu\neq\mu_{0}$	$\|t\|>1.96$ no such rule
Example: $t=2.10$ : this rule wrongly rejects, $2.10<2.262$ . What to do: With $n<30$ , the critical value is read from the Student table, with $n-1$ degrees of freedom.

At $1\%$ : $2.576$ two-tailed, $2.326$ one-tailed. The smaller the number of degrees of freedom, the further Student's critical value moves from that of the normal distribution.

The mistakes that cost marks

These are the errors I correct most often in session. Each one costs marks on a paper, even when the reasoning behind it is right.

1. Writing the hypotheses about the sample mean

1 mark, and the conclusion loses its object

What not to write

“ $H_{0}:\overline{x}=120$ s and $H_{1}:\overline{x}<120$ s.”

What to write

“ $H_{0}:\mu=120$ s and $H_{1}:\mu<120$ s, where $\mu$ is the mean reaction time with the catalyst.”

Why: $\overline{x}$ is known as soon as the trials are done: we do not make a hypothesis about a calculated number. The test is about the unknown parameter of the population.

2. Concluding that the null hypothesis is true because it is not rejected

1 to 2 marks, the whole conclusion

What not to write

“We do not reject H₀: the catalyst has no effect on the reaction time.”

What to write

“At the 5% level, the data do not allow us to conclude that the catalyst reduces the mean reaction time.”

If the true mean is

117

\overline{x}

stays above the bound

116.1

in about

65\%

of cases (blue area): not rejecting is then the most likely outcome.

Why: A small real gap often goes unnoticed. With $n=40$ trials and $s=15$ s, a catalyst that brings the mean down to $117$ s escapes the test in about $65\%$ of cases.

3. Comparing with 1.96 in a one-tailed test

2 marks: the decision and the conclusion are wrong

What not to write

“Right-tailed test at the $5\%$ level: $z=1.80<1.96$ , so we do not reject $H_{0}$ .”

What to write

“Right-tailed: we reject if $z>1.645$ . Here $z=1.80>1.645$ , so we reject $H_{0}$ at the $5\%$ level.”

The same statistic

z=1.80

falls in the rejection zone of the one-tailed test (top) but not in that of the

1.96

rule, which belongs to the two-tailed test (bottom).

Why: $1.96$ leaves $2.5\%$ in EACH tail: it is the two-tailed value. In a one-tailed test, the $5\%$ are all in a single tail, beyond $1.645$ .

4. Reading the p-value as the probability that the null hypothesis is true

1 interpretation mark

What not to write

“ $p=0.03$ : there is a $3\%$ chance that the fertilizer has no effect.”

What to write

“If the fertilizer had no effect, a mean yield at least this high would appear in only $3\%$ of samples.”

Why: The $p$ -value is computed assuming $H_{0}$ is true: it is $P(\text{data at least this extreme}\mid H_{0})$ , not $P(H_{0}\mid\text{data})$ . The two differ as in Bayes' theorem.

5. Putting the observed proportion in the standard error of a test

2 marks: here the decision flips

What not to write

“ $\hat{p}=\frac{84}{240}=0.35$ and $z=\dfrac{0.35-0.30}{\sqrt{0.35\times 0.65/240}}\approx 1.62<1.645$ : we do not reject.”

What to write

“Under $H_{0}$ , $\sigma_{\hat{P}}=\sqrt{\frac{0.30\times 0.70}{240}}\approx 0.0296$ and $z\approx 1.69>1.645$ : we reject $H_{0}$ at the $5\%$ level.”

Why: The test reasons under $H_{0}$ , hence with $p=p_{0}$ . It is the confidence interval, which has no assumed value, that uses $\hat{p}$ .

6. Using the normal distribution with a small sample

2 marks: wrong distribution, wrong decision

What not to write

“ $n=5$ , $t=2.40>1.96$ : we reject $H_{0}$ at the $5\%$ level.”

What to write

“ $n=5$ : Student with $4$ degrees of freedom, critical value $2.776$ . $|t|=2.40<2.776$ , so we do not reject $H_{0}$ .”

Why: With few data, $s$ is a poor estimate of $\sigma$ : Student's t distribution has heavier tails, and its critical value is larger. Comparing with $1.96$ takes a risk well above $5\%$ .

7. Choosing the alternative hypothesis after seeing the data

1 mark, and a real risk that is doubled

What not to write

“The question asks whether the measurement of $g$ is biased. Since $\overline{x}<9.81$ , I test $H_{1}:\mu<9.81$ .”

What to write

“The question is about a bias in either direction: $H_{1}:\mu\neq 9.81$ , a two-tailed test, chosen before computing $\overline{x}$ .”

Why: Choosing the side afterwards amounts to keeping $5\%$ in the tail where the data fall AND the possibility of the other one: the true Type I error risk rises to $10\%$ .

8. Forgetting to double the p-value of a two-tailed test

1 mark, and a wrong decision at the 2% level

What not to write

“Two-tailed test, $z=2.10$ : $p=P(Z>2.10)=0.0179$ .”

What to write

“Two-tailed test: $p=2\times P(Z>2.10)=2\times 0.0179=0.0358$ .”

Why: In a two-tailed test, a gap on the other side would have been just as surprising: both tails count. At the $2\%$ level, $0.0179$ would lead to rejecting, $0.0358$ would not.

9. Concluding without the context

1 mark, often the last one of the question

What not to write

“z = −2.20 < −1.645, so we reject H₀.”

What to write

“z = −2.20 < −1.645: we reject H₀ at the 5% level. The data allow us to conclude that the mean lead concentration of the network is below the standard.”

Why: The decision is a statistical step; the conclusion is a scientific sentence, with the parameter named and the level stated. The marker grades the two separately.

Which method to choose

Which test statistic?

Look at what is tested (a mean or a proportion), then at the sample size and what is known about the population.

With

4

degrees of freedom, Student's t distribution is more spread out than the normal distribution: the two-tailed critical value at

5\%

goes from

1.96

2.776

If a mean, $n\geq 30$ → $z=\dfrac{\overline{x}-\mu_{0}}{s/\sqrt{n}}$ , normal distribution

Example: $n=40$ , $s=0.52$ : standard error $0.082$

the central limit theorem means the population need not be assumed normal
If a mean, $n<30$ , population assumed normal → $t$ , Student's t distribution with $n-1$ degrees of freedom

Example: $n=12$ : $t_{0.05,\,11}=1.796$ instead of $1.645$
If a mean, $n<30$ , strongly skewed data or an outlier → no test of the course applies: more data are needed

say so on the paper instead of forcing a calculation
If a proportion, $np_{0}\geq 5$ and $n(1-p_{0})\geq 5$ → $z=\dfrac{\hat{p}-p_{0}}{\sqrt{p_{0}(1-p_{0})/n}}$ , normal distribution

Example: $84$ out of $240$ against $p_{0}=0.30$ : $z\approx 1.69$

The choice is made from the statement, before any calculation, and the condition that justifies it is written on the paper.

One-tailed or two-tailed: read the verb of the question

Look in the statement for what the study wants to establish, without looking at the data.

If “has changed”, “differs”, “is biased”, “matches the stated value” → two-tailed: $H_{1}:\mu\neq\mu_{0}$

Example: measurement of $g$ compared with $9.81$ m/s²
If “increases”, “exceeds”, “is higher” → right-tailed: $H_{1}:\mu>\mu_{0}$

Example: a fertilizer that increases the yield
If “decreases”, “reduces”, “is lower than” → left-tailed: $H_{1}:\mu<\mu_{0}$

Example: a catalyst that reduces the reaction time
If compliance with a maximum standard must be PROVED → $H_{1}$ : below the standard, $\mu<\mu_{0}$

Example: water declared compliant only if the proof is made

the most serious error, declaring unsafe water safe, becomes the one the level controls

If the statement says nothing about the direction, the test is two-tailed.

How the answer is expected to be written

A marker ticks steps. Here they are in order, with the concluding sentence expected word for word.

Writing up a complete test

When to use it: The statement asks you to test, to check a claim, or to say whether the data allow a conclusion.

1 Name the parameter in words, then write $H_{0}$ (the equality) and $H_{1}$ (what we want to establish), and the level $\alpha$ .
2 Write the conditions and the distribution chosen: $n\geq 30$ or normal population, $np_{0}\geq 5$ , random sample.
3 Compute the statistic, showing the standard error used (with $s$ , or with $p_{0}$ ).
4 Give the critical value (or the $p$ -value) and compare.
5 Decide (reject or do not reject $H_{0}$ ), then conclude in context, with the level.

Concluding sentence

“ $t\approx 2.60>1.796$ : we reject $H_{0}$ at the $5\%$ level. The data allow us to conclude that the fertilizer increases the mean yield of the variety.”

The trap: Skipping the conditions: the calculation is right, but nothing shows that it was allowed.

Marking: Typical breakdown: hypotheses, conditions and distribution, statistic, decision, conclusion in context, in roughly equal parts.

Concluding when you do not reject

When to use it: The statistic does not enter the critical region, or the p-value exceeds the level.

1 Write the comparison: “ $z=-1.45>-1.645$ ” or “ $p=0.07>0.05$ ”.
2 Decide: “we do not reject $H_{0}$ at the $5\%$ level”.
3 Conclude on what the data do NOT allow you to establish, with the vocabulary of the statement.
4 If the question asks for it, name the limitation: small sample, low power.

Concluding sentence

“At the 5% level, the data do not allow us to conclude that the mean concentration exceeds 20 µg/L.”

The trap: Writing “we accept H₀” or “the concentration is 20 µg/L”: the test never proves the null hypothesis.

Check before you hand in

Five minutes of checking recover more marks than one more problem started in a hurry.

The sign of the statistic and the direction of H₁

In a one-tailed test, you can reject only if the statistic is on the side of $H_{1}$ : positive for $\mu>\mu_{0}$ , negative for $\mu<\mu_{0}$ .

$H_{1}:\mu>3.20$ and $z=-0.8$ : rejection is impossible, whatever the level; if your calculation concludes to reject, the subtraction is the wrong way round.
The p-value and the critical region say the same thing

$p\leq\alpha$ exactly when the statistic is in the critical region. Check that the two readings agree.

$z=1.80$ in a one-tailed test and $p=0.0359$ : $p<0.05$ and $z>1.645$ , consistent. A $p$ -value of $0.07$ with a rejection reveals a doubling that does not belong there, or a halving that was forgotten.
The order of magnitude in standard errors

Compare $|\overline{x}-\mu_{0}|$ with the standard error: below one standard error, no test rejects at the usual levels; beyond three, all reject at $5\%$ , except with Student and very few data.

A gap of $0.18$ for a standard error of $0.082$ : a little over two standard errors, $z\approx 2.2$ . A $z$ of $0.35$ signals a division by $s$ instead of $s/\sqrt{n}$ .
The degrees of freedom

Student is read with $n-1$ degrees of freedom, and its critical value is always larger than that of the normal distribution at the same level.

$n=12$ : row $11$ of the table, $1.796>1.645$ . A Student critical value below $1.645$ in a one-tailed test at $5\%$ is a reading error.
The confidence interval confirms

In a two-tailed test at level $\alpha$ , we reject exactly when $\mu_{0}$ falls outside the $1-\alpha$ confidence interval (same standard error).

Interval $[9.748,\,9.800]$ and $\mu_{0}=9.81$ : outside the interval, the test must reject. If it does not, one of the two calculations is wrong.

The typical problem, taken apart

A fertilizer trial on twelve plots

Without fertilizer, the mean yield of a variety is $3.20$ t/ha. A trial on $n=12$ plots chosen at random gives $\overline{x}=3.47$ t/ha and $s=0.36$ t/ha. The yields are assumed to be normally distributed.

Does the fertilizer increase the mean yield, at the $5\%$ level? We give $t_{0.05,\,11}=1.796$ , $t_{0.025,\,11}=2.201$ and $t_{0.01,\,11}=2.718$ .

Step 1

$\mu$ : mean yield of the variety with fertilizer. $H_{0}:\mu=3.20$ and $H_{1}:\mu>3.20$ , at level $\alpha=0.05$ .

Why

The word “increase” fixes a right-tailed test, before looking at the observed mean.

Step 2

$n=12<30$ , $\sigma$ unknown, population assumed normal: Student's t distribution with $11$ degrees of freedom.

Why

This line earns the conditions mark; without it, nothing justifies the table used.

Step 3

$\dfrac{s}{\sqrt{n}}=\dfrac{0.36}{\sqrt{12}}\approx 0.1039$ and $t=\dfrac{3.47-3.20}{0.1039}\approx 2.60$

Why

We measure the gap from H₀ in standard errors: 0.27 t/ha is a little more than two and a half standard errors.

Step 4

Right-tailed: we reject if $t>1.796$ . Here $2.60>1.796$ : we reject $H_{0}$ .

Why

The critical value comes from row 11 of the table, not 1.645: with twelve plots, the normal distribution would underestimate the risk.

Step 5

$2.201<2.60<2.718$ , so $0.01<p<0.025$ .

Why

The bracket on the p-value tells the strength of the result: rejection at 5% and at 2.5%, not at 1%.

Step 6

Check: $95\%$ lower confidence bound, $3.47-1.796\times 0.1039\approx 3.28>3.20$ .

Why

A one-sided bound above 3.20 gives the same decision as the one-tailed test: two calculations, a single conclusion.

The conclusion, written out

“ $t\approx 2.60>1.796$ : we reject $H_{0}$ at the $5\%$ level. The trial data allow us to conclude that the fertilizer increases the mean yield of this variety ( $0.01<p<0.025$ ).”

The classic mistake on this problem: Comparing $2.60$ with $1.645$ , the normal value, or reading the table at $12$ degrees of freedom instead of $11$ . Here the decision does not change, but the method loses its marks; with $t=1.70$ , the normal distribution would have led to a wrong rejection, since $1.645<1.70<1.796$ .

Learn by heart

• The hypotheses are about $\mu$ or $p$ , never about $\overline{x}$ or $\hat{p}$ ; $H_{0}$ carries the EQUALITY.
• Critical values of the normal distribution: $1.96$ (two-tailed at $5\%$ ), $1.645$ (one-tailed at $5\%$ ), $2.576$ and $2.326$ at $1\%$ .
• Mean with $n<30$ : Student with $n-1$ degrees of freedom, population assumed normal.
• Test on a proportion: $p_{0}$ in the standard error AND in the conditions $np_{0}\geq 5$ , $n(1-p_{0})\geq 5$ .
• $p$ -value: the probability, IF $H_{0}$ is true, of a result at least as extreme; reject if $p\leq\alpha$ .
• $\alpha=P(\text{rejecting }H_{0}\mid H_{0}\text{ true})$ ; $\beta=P(\text{not rejecting}\mid H_{0}\text{ false})$ ; power $=1-\beta$ .
• Not rejecting H₀ NEVER proves it: the conclusion is that the data do not allow a conclusion.

Frequently asked questions

How do you know whether a hypothesis test is one-tailed or two-tailed?

You decide from the question asked, before looking at the data. If the study wants to establish a specific increase or decrease, with words such as increases, exceeds, reduces or is lower, the test is one-tailed on the side of that word. If it only asks whether the value has changed or differs, in either direction, it is two-tailed. With no indication of direction, take the two-tailed test.

When should you use Student's t distribution rather than the normal distribution in a test on a mean?

When the sample has fewer than 30 observations and the population standard deviation is unknown, you use Student's t distribution with n minus 1 degrees of freedom, provided the population is assumed normal. With 30 observations or more, the central limit theorem allows the normal distribution, even if the data are not normal. Student's critical value is always larger than that of the normal distribution.

What does a p-value of 0.03 mean?

It means that, if the null hypothesis were true, a result at least as far from that hypothesis as the one observed would occur in only 3 percent of samples. So the null hypothesis is rejected at the 5 percent level, but not at the 1 percent level. It is not the probability that the null hypothesis is true.

Can you accept the null hypothesis when the test does not reject it?

No. Not rejecting the null hypothesis only means that the data are not far enough from it to rule out chance. A real but small effect, or a sample that is too small, often gives this result. So you write that the data do not allow you to conclude, at the chosen level, what the study wanted to establish.

Mark this sheet as read or add it to your favourites: a free account, no password, keeps your read sheets and favourites from one visit to the next, tells you which chapter to tackle next and lets you ask for the method or exercise you are missing. Create your space, an email is enough.

Practise it

Corrected exercises: Hypothesis tests, 201-SN1-RE

A method is proved on a paper, not on a sheet. The set for the same chapter takes each of these traps into a problem, with the solution written out step by step.

11 corrected exercises
110 points
180 minutes

Do the exercises

Something missing from this sheet? A method, a trap, an exercise to practise it, a mistake you spotted: tell me, I read every request.

Sending goes through your free space: that is how I can answer you and tell you when the exercise is added. An email is enough, no password. Create your space. Rather not create an account? Write to me directly.

Previous sheet Confidence intervals Next sheet Regression and the chi-square test

This chapter comes back in

The chapters that build on it, later in the year or in the years that follow.

Linear regression and the chi-square test of independence

© Ahmed Squalli Houssaini. Revision sheet published at www.letuteurscientifique.ca/en/fiches/cegep-sn1-hypothesis-tests. Free for personal and classroom use; republishing it elsewhere requires written permission (legal notice).

Revision sheet: hypothesis tests (201-SN1-RE)

The essentials

The logic of a test: reasoning under the null hypothesis

The three statistics of the course, and their conditions

The two errors, and what controls them

The rules in table form

When to reject $H_{0}$ at the $5\%$ level

The mistakes that cost marks

1. Writing the hypotheses about the sample mean

2. Concluding that the null hypothesis is true because it is not rejected

3. Comparing with 1.96 in a one-tailed test

4. Reading the p-value as the probability that the null hypothesis is true

5. Putting the observed proportion in the standard error of a test

6. Using the normal distribution with a small sample

7. Choosing the alternative hypothesis after seeing the data

8. Forgetting to double the p-value of a two-tailed test

9. Concluding without the context

Which method to choose

Which test statistic?

One-tailed or two-tailed: read the verb of the question

How the answer is expected to be written

Writing up a complete test

Concluding when you do not reject

Check before you hand in

The typical problem, taken apart

A fertilizer trial on twelve plots

Learn by heart

Frequently asked questions

How do you know whether a hypothesis test is one-tailed or two-tailed?

When should you use Student's t distribution rather than the normal distribution in a test on a mean?

What does a p-value of 0.03 mean?

Can you accept the null hypothesis when the test does not reject it?

Corrected exercises: Hypothesis tests, 201-SN1-RE

This chapter comes back in

See also

Looking for a 201-SN1-RE tutor in Montreal?

Revision sheet: hypothesis tests (201-SN1-RE)

The essentials

The logic of a test: reasoning under the null hypothesis

The three statistics of the course, and their conditions

The two errors, and what controls them

The rules in table form

When to reject H0H_{0}H0​ at the 5%5\%5% level

The mistakes that cost marks

1. Writing the hypotheses about the sample mean

2. Concluding that the null hypothesis is true because it is not rejected

3. Comparing with 1.96 in a one-tailed test

4. Reading the p-value as the probability that the null hypothesis is true

5. Putting the observed proportion in the standard error of a test

6. Using the normal distribution with a small sample

7. Choosing the alternative hypothesis after seeing the data

8. Forgetting to double the p-value of a two-tailed test

9. Concluding without the context

Which method to choose

Which test statistic?

One-tailed or two-tailed: read the verb of the question

How the answer is expected to be written

Writing up a complete test

Concluding when you do not reject

Check before you hand in

The typical problem, taken apart

A fertilizer trial on twelve plots

Learn by heart

Frequently asked questions

How do you know whether a hypothesis test is one-tailed or two-tailed?

When should you use Student's t distribution rather than the normal distribution in a test on a mean?

What does a p-value of 0.03 mean?

Can you accept the null hypothesis when the test does not reject it?

Corrected exercises: Hypothesis tests, 201-SN1-RE

This chapter comes back in

See also

Looking for a 201-SN1-RE tutor in Montreal?

When to reject $H_{0}$ at the $5\%$ level