Hypothesis tests & confidence intervals: practice problems with step-by-step solutions

Also called: t-test · p-value

Free Hypothesis tests & confidence intervals practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: z statistic, t statistic, confidence intervals.

Free practice problemEasy
μ₀x̄c = ?α = 0.05

A supplier claims that the mean breaking strength of a cable is 50 kN; you suspect it is higher. The standard deviation is known to be 8 kN. You will take a sample of 25 cables and test H₀: μ = 50 kN against H₁: μ > 50 kN at α = 0.05. Above which sample mean x̄c will you reject H₀?

μ₀ = 50 kNσ = 8 kNn = 25α = 0.05
kN
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What you practise

  1. Easyz statistic
  2. Mediumt statistic
  3. Hardconfidence intervals

How to solve Hypothesis tests & confidence intervals problems

  1. State H₀ and H₁ and decide whether the test is one- or two-sided.
  2. Calculate the test statistic: (x̄ − μ₀)/(σ/√n), or with s and t.
  3. Compare with the critical value, or find the p-value.
  4. Reject H₀ if the statistic is beyond the critical value (p below α), and state the conclusion in words.

Common mistakes

  • Using a one-sided critical value for a two-sided test.
  • Saying H₀ is proven true when it is not rejected.
  • Forgetting √n in the standard error.

Key formulas

z-test for a mean
z = (x̄ − μ0)/(σ/√n)
t-test for a mean
t = (x̄ − μ0)/(s/√n), df = n − 1
Two-sided at 5%
reject H0 if |z| > 1.96
Confidence interval
x̄ ± z·σ/√n

Learn Hypothesis tests & confidence intervals from scratch

Before this: Confidence intervals

1The idea of a test

A machine should fill bottles with 500 mL on average. A sample of bottles has a mean of 496 mL. Is the machine wrong, or is this just chance?

A hypothesis test answers that question with numbers. You start from two statements:

  • The null hypothesis H₀: nothing has changed. Here H₀: μ = 500.
  • The alternative hypothesis H₁: what you suspect. Here H₁: μ ≠ 500 (two-sided), or μ < 500 if you only worry about underfilling (one-sided).

You assume H₀ is true and ask: how unusual is this sample? If it would be very unusual, you reject H₀.

Self-check A supplier claims that its resistors are 220 Ω on average. You want to test this claim. What is H₀?
  1. x̄ = 220 Ω
  2. μ > 220 Ω
  3. μ = 220 Ω
  4. μ ≠ 220 Ω

Answer: μ = 220 Ω. H₀ is the claim being tested, about the population mean μ. μ ≠ 220 Ω is the alternative H₁, and a hypothesis is never about the sample mean x̄, which you already know.

2The test statistic

The test statistic measures how far the sample mean lies from μ₀, the value in H₀, in units of the standard error.

z = (x̄ − μ₀) / (σ / √n)

Use z when the population standard deviation σ is known. When only the sample standard deviation s is known, use t:

t = (x̄ − μ₀) / (s / √n), df = n − 1

A value of z or t near 0 means the sample agrees with H₀. A large positive or negative value means the sample is far from μ₀.

Self-check H₀: μ = 120, σ = 8, n = 16 and x̄ = 118. What is z?
  1. 1
  2. −0.25
  3. −1
  4. −4

Answer: −1. The standard error is 8/√16 = 2, so z = (118 − 120)/2 = −1. Dividing by σ = 8 instead of σ/√n gives −0.25. The sign is negative because x̄ is below μ₀.

3Deciding: critical values and p-values

Before you look at the data, choose the significance level α, usually 0.05. It is the risk you accept of rejecting a true H₀.

For a z test at α = 0.05:

  • Two-sided (H₁: μ ≠ μ₀): reject H₀ if |z| > 1.96.
  • One-sided (H₁: μ > μ₀): reject H₀ if z > 1.645. For μ < μ₀: reject if z < −1.645.

The p-value is the probability of a result at least this extreme if H₀ is true. Reject H₀ when p < α.

Not rejecting H₀ does not prove it. It only means the data are not strong enough evidence against it.

A test and a confidence interval agree: a two-sided test at 5% rejects H₀ exactly when μ₀ lies outside the 95% interval x̄ ± 1.96 · σ/√n.

Self-check A two-sided z test at α = 0.05 gives z = 1.8. What do you conclude?
  1. Do not reject H₀
  2. H₀ is proven true
  3. Reject H₀
  4. Reject H₁ and accept H₀ as proven

Answer: Do not reject H₀. |1.8| is smaller than 1.96, so the result is not significant and H₀ is not rejected. That does not prove H₀; the evidence is just not strong enough.

4The method step by step

  1. State H₀ and H₁, and decide whether the test is one- or two-sided.
  2. Choose z (σ known) or t (only s known, df = n − 1).
  3. Compute the standard error σ/√n or s/√n.
  4. Compute the test statistic (x̄ − μ₀) / standard error.
  5. Compare with the critical value, or find the p-value.
  6. State the conclusion in words, in terms of the question.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A machine should fill bottles with 500 mL on average. σ = 12 mL. A sample of 36 bottles has a mean of 496 mL. Test H₀: μ = 500 against H₁: μ ≠ 500 at α = 0.05.

1

Hypotheses

H₀: μ = 500 mL. H₁: μ ≠ 500 mL. This is two-sided, so the critical values are ±1.96.

2

Standard error

σ is known, so use z.

σ / √n = 12 / √36 = 12 / 6 = 2 mL
3

Test statistic

z = (496 − 500) / 2 = −4 / 2 = −2
4

Compare

−1.9601.962.5%2.5%95%z = −2

|−2| = 2 is larger than 1.96, so z lies in the shaded rejection region.

5

Check with an interval

496 ± 1.96 · 2 = [492.08, 499.92]

The value 500 lies outside the 95% interval, so the interval gives the same decision.

Answer

z = −2. Reject H₀ at the 5% level: the machine’s mean fill is different from 500 mL.

6Common mistakes

  • Using a one-sided critical value (1.645) for a two-sided test, or the other way round.
  • Saying H₀ is proven true when it is not rejected.
  • Forgetting √n in the standard error.
  • Writing x̄ − μ₀ the wrong way round, which flips the sign of z.
  • Using z when only the sample s is known for a small sample.
Self-check A one-sided test with H₁: μ > μ₀ at α = 0.05 gives z = 1.8. What do you conclude?
  1. Do not reject H₀, because 1.8 < 1.96
  2. Do not reject H₀, because 1.8 < 2.576
  3. Reject H₀, because 1.8 > 1.645
  4. Reject H₀, because z is positive

Answer: Reject H₀, because 1.8 > 1.645. One-sided at 5% puts the whole 5% in one tail, so the critical value is 1.645. Comparing with 1.96 uses the two-sided value by mistake.

7How exam questions look

  • Easy: the z statistic. σ is known; compute z = (x̄ − μ₀)/(σ/√n).
  • Medium: the t statistic. Only the sample standard deviation s is given; compute t = (x̄ − μ₀)/(s/√n).
  • Hard: confidence intervals. Find the lower or upper limit of x̄ ± z · σ/√n at 95% or 99%.

A quick check: if x̄ is below μ₀, the test statistic must be negative.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
z −1.9601.96

A machine is supposed to fill bottles with 100 mL on average. The standard deviation is known to be 4 mL. A sample of 81 bottles has a mean of 99.6 mL. Find the test statistic z for H₀: μ = 100.

μ₀ = 100 mLx̄ = 99.6 mLσ = 4 mLn = 81

Find z

Show hints
  1. The standard error of the mean is the standard deviation divided by √n.
  2. SE = 4/√81 = 0.4444.
  3. Then divide the difference x̄ − μ₀ by the standard error. Keep the sign.
Show the answer and the step-by-step solution

Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).

Standard error

SE = 4/√81 = 4/9 = 0.4444

Test statistic

z = (99.6 − 100)/0.4444 = −0.9

Answer

z = −0.9

|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).

Example 2Medium
t −1.99801.998

A sample of 64 students has a mean score of 504.5 and a sample standard deviation of 15. Find the test statistic t for H₀: μ = 500.

μ₀ = 500x̄ = 504.5s = 15n = 64

Find t

Show hints
  1. The standard error of the mean is the standard deviation divided by √n.
  2. SE = 15/√64 = 1.875.
  3. Then divide the difference x̄ − μ₀ by the standard error. Keep the sign.
Show the answer and the step-by-step solution

Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).

Standard error

SE = 15/√64 = 15/8 = 1.875

Test statistic

t = (504.5 − 500)/1.875 = 2.4

Answer

t = 2.4
t0.025; df = 63 = 1.998

|t| > 1.998, so H₀ is rejected at the 5% level (two-sided).

Example 3Hard
z −2.57602.576

A sample of 64 measurements has mean 80. The population standard deviation is 20. Find the lower limit L of the 99% confidence interval for the mean.

x̄ = 80σ = 20n = 6499%

Find L

Show hints
  1. Confidence interval: x̄ ± z*·σ/√n.
  2. z* = 2.576 for 99%.
  3. σ/√n = 2.5.
Show the answer and the step-by-step solution

Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).

Margin of error

E = 2.576·20/√64 = 6.44

Interval limit

L = 80 − 6.44 = 73.56

Answer

L = 73.56

Questions and answers

What is Hypothesis tests & confidence intervals?

A machine should fill bottles with 500 mL on average. A sample of bottles has a mean of 496 mL. Is the machine wrong, or is this just chance?

A hypothesis test answers that question with numbers. You start from two statements:

How do I solve Hypothesis tests & confidence intervals problems step by step?
  1. State H₀ and H₁ and decide whether the test is one- or two-sided.
  2. Calculate the test statistic: (x̄ − μ₀)/(σ/√n), or with s and t.
  3. Compare with the critical value, or find the p-value.
  4. Reject H₀ if the statistic is beyond the critical value (p below α), and state the conclusion in words.
What are common mistakes in Hypothesis tests & confidence intervals?
  • Using a one-sided critical value for a two-sided test.
  • Saying H₀ is proven true when it is not rejected.
  • Forgetting √n in the standard error.
What do the Hypothesis tests & confidence intervals exercises cover?
  • Easy: z statistic
  • Medium: t statistic
  • Hard: confidence intervals

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

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Printable worksheet

Hypothesis tests & confidence intervals worksheet with answers
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