ETX5900 · Business Statistics
Hypothesis Testing for Business Decisions I
Module 7 (Week 7) of ETX5900 Business Statistics at Monash University introduces formal hypothesis testing — the procedure that turns a business claim about a population into a decision under uncertainty. You learn the five-step framework, the one-sample test of a mean in its two forms (a Z-test when the population standard deviation is known and a Student-t test when it is unknown), one- versus two-tailed reject rules, and the Type I / Type II error pair. It maps to Berenson (Australian 5th ed.) Chapter 9 and sets up the proportion and two-sample tests of the following module.
What this chapter covers
- 01The five-step framework: state H0 and H1, compute the test statistic, find the critical value or p-value at level alpha, decide, conclude in context
- 02H0 (null) always contains the equality; H1 (alternative) carries the research claim and sets the tail
- 03One-sample mean, sigma KNOWN: Z = (xbar - mu0) / (sigma / sqrt(n)), compared to the standard normal
- 04One-sample mean, sigma UNKNOWN: t = (xbar - mu0) / (s / sqrt(n)) with df = n - 1, compared to Student-t
- 05Two-tailed (H1: mu != mu0): reject if |statistic| > critical value at alpha/2 in each tail
- 06One-tailed (H1: mu > mu0 or mu < mu0): reject if the statistic falls in the single tail beyond the alpha critical value
- 07The p-value method: reject H0 if p-value < alpha, otherwise do not reject
- 08Type I error = reject a true H0 (probability alpha); Type II error = fail to reject a false H0 (probability beta)
- 09Power = 1 - beta; for a fixed n, lowering alpha raises beta (an inverse trade-off) and only a larger n reduces both
- 10We never 'accept H0' or 'prove H1' - we reject or fail to reject H0
Two-tailed t-test of a mean (sigma unknown): battery life against a 500-hour claim
- +1Step 1 - hypotheses. The claim is 'differs from', so the test is two-tailed: H0: mu = 500 versus H1: mu != 500.
- +1Step 2 - choose the statistic. The population sigma is unknown, so use the Student-t statistic with df = n - 1 = 15, not the Z-test.
- +1Step 2 (cont.) - compute it. Standard error = s / sqrt(n) = 20 / sqrt(16) = 20 / 4 = 5. t = (xbar - mu0) / (s / sqrt(n)) = (486 - 500) / 5 = -14 / 5 = -2.80.
- +1Step 3 - decision rule. Two-tailed at alpha = 0.05 puts alpha/2 = 0.025 in each tail. From the t-table, t(15, 0.025) = 2.131; reject H0 if |t| > 2.131.
- +1Step 4 - decide. |t| = 2.80 > 2.131, so the statistic lands in the rejection region: reject H0. (Equivalently the two-tailed p-value is about 0.013, which is less than 0.05.)
- +1Step 5 - conclude in context. There is sufficient evidence at the 5% level that the true mean battery life differs from 500 hours; because t is negative and xbar (486) is below 500, the batteries appear to last less than claimed.
Key terms
- Null hypothesis (H0)
- The status-quo or 'no effect' claim being tested; it always contains the equality (=, <= or >=). A test can only reject or fail to reject H0 - never 'accept' or 'prove' it.
- Alternative hypothesis (H1)
- The research claim you are testing for. Its form sets the tail: != gives a two-tailed test, > an upper one-tailed test, and < a lower one-tailed test.
- Test statistic
- A standardised measure of how far the sample result sits from the value in H0. For one mean it is Z = (xbar - mu0)/(sigma/sqrt(n)) when sigma is known, or t = (xbar - mu0)/(s/sqrt(n)), df = n - 1, when sigma is unknown.
- Significance level (alpha)
- The chosen probability of a Type I error, e.g. 0.05. It sets the size of the rejection region and, for a p-value test, the cut-off: reject H0 if p-value < alpha.
- Critical value
- The table cut-off marking the edge of the rejection region: z(alpha/2) or t(n-1, alpha/2) for a two-tailed test, z(alpha) or t(n-1, alpha) for a one-tailed test. Reject H0 if the statistic is more extreme than this value.
- p-value
- The probability, assuming H0 is true, of a test statistic at least as extreme as the one observed (both tails for a two-tailed test). Universal rule: reject H0 if p-value < alpha, otherwise do not reject.
- Type I error (alpha)
- Rejecting H0 when H0 is actually true - a 'false alarm'. Its probability equals the significance level alpha, which you choose in advance.
- Type II error (beta) and power
- A Type II error is failing to reject H0 when it is false - a 'miss', with probability beta. Power = 1 - beta is the chance of correctly rejecting a false H0; for fixed n, a smaller alpha raises beta, and only a larger sample lowers both.
Hypothesis Testing for Business Decisions I FAQ
When do I use the Z-test and when the t-test for a single mean?
It depends on whether the population standard deviation sigma is known. If sigma is known, use Z = (xbar - mu0)/(sigma/sqrt(n)) and the standard normal table. If sigma is unknown and you estimate it with the sample standard deviation s, use t = (xbar - mu0)/(s/sqrt(n)) with df = n - 1 and the t-table. In real business data sigma is almost never known, so the t-test is the usual choice; the t curve has heavier tails, giving slightly larger critical values to reflect the extra uncertainty.
What is the difference between a Type I and a Type II error?
A Type I error is rejecting a true null hypothesis - a false alarm - and its probability is the significance level alpha that you set (e.g. 0.05). A Type II error is failing to reject a false null hypothesis - a miss - with probability beta, which you do not choose directly; it depends on the true effect size, the sample size and alpha. For a fixed sample size the two trade off: making alpha smaller reduces Type I errors but raises beta, so only increasing the sample size lowers both at once.
Can AI help me with hypothesis testing in ETX5900?
Yes — as a study aid. Sia (AskSia) walks through the five-step framework step by step — stating H₀ and H₁, choosing the Z- or t-test, picking one- versus two-tailed, reading the critical value from the tables, and phrasing the reject / do-not-reject conclusion in context — generates fresh practice problems at the level you need, and checks your own working so you learn the method. It never hands over answers, sits an assessment in your place, or guarantees a grade — always follow Monash's academic-integrity rules and confirm what is permitted on Moodle.
Exam move
Treat hypothesis testing as a fixed procedure rather than a memory test: the exam provides a formula sheet and statistical tables, so marks come from applying the five steps cleanly, not from recall. Drill the decision points until they are automatic - is sigma known (Z) or unknown (t, df = n - 1)? is the claim 'differs' (two-tailed, alpha/2 in each tail) or directional (one-tailed, all alpha in one tail)? does the statistic fall beyond the critical value, or is the p-value below alpha? Always write H0 with the equality, phrase the conclusion about H1 in business language, and say 'do not reject' rather than 'accept'. Because the Final Examination is worth 50% and its duration is not stated in the unit materials, plan by marks rather than minutes: spend time on each question in proportion to its marks and show the statistic before the decision so you earn method marks. Confirm the exam date, length and format on Moodle / my.Monash.
Working through Hypothesis Testing for Business Decisions I in ETX5900? Sia is AskSia’s AI Statistics tutor — ask any ETX5900 Hypothesis Testing for Business Decisions I question and get a clear, step-by-step explanation grounded in how ETX5900 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
