ETX5900 · Business Statistics
Hypothesis Testing for Business Decisions II
Module 8 (Week 8) of ETX5900 Business Statistics at Monash University extends the hypothesis-testing frame from a single mean to a single proportion and to two-sample comparisons, and introduces R as an analysis tool. You test claims about a population proportion with a z-statistic whose standard error is built from the hypothesised value, then compare two groups (two proportions with a pooled z, two means with a t). It maps to Berenson (Australian 5th ed.) Chapters 9-10 and builds directly on the normal / sampling-distribution and confidence-interval chapters.
What this chapter covers
- 01The five-step testing frame: hypotheses, test statistic, critical value / p-value, decision, conclusion in context
- 02H0 always contains '='; H1 is the research claim and sets the tail (one- vs two-tailed)
- 03One-sample proportion z-test: Z = (p - pi0) / sqrt(pi0(1 - pi0) / n), with the SE built from the hypothesised pi0
- 04Validity condition for the normal approximation: n*pi0 >= 5 and n(1 - pi0) >= 5
- 05Reject rules by tail: |z| > z(alpha/2) two-tailed, z beyond z(alpha) one-tailed, or p-value < alpha
- 06Two-sample difference of proportions: pooled p-bar = (X1 + X2)/(n1 + n2) in a z-statistic
- 07Two-sample difference of means with sigma unknown: a t-statistic (exact SE and df are on the exam formula sheet)
- 08Type I error (probability alpha) vs Type II error (probability beta), and power = 1 - beta
- 09Never 'accept H0' - we reject or fail to reject; a non-significant result is 'insufficient evidence'
- 10Running these tests in R with prop.test() (set correct = FALSE to match the textbook z-test)
One-sample proportion test: do more than 60% of graduates find a job within three months?
- +1State the hypotheses. The claim 'more than 60%' is the research claim, so it is the alternative: H0: pi = 0.60 versus H1: pi > 0.60. Because H1 uses '>', this is a one-tailed (upper) test.
- +1Sample proportion and validity. p = X/n = 126/180 = 0.70. Check the normal approximation: n*pi0 = 180 x 0.60 = 108 >= 5 and n(1 - pi0) = 180 x 0.40 = 72 >= 5, so the z-test is valid.
- +1Standard error, built from the hypothesised pi0: SE = sqrt(pi0(1 - pi0)/n) = sqrt(0.60 x 0.40 / 180) = sqrt(0.0013333) = 0.036515.
- +1Test statistic: z = (p - pi0)/SE = (0.70 - 0.60)/0.036515 = 0.10/0.036515 = 2.74.
- +1Critical value / p-value. One-tailed critical value z(0.05) = 1.645. p-value = P(Z > 2.74) = 1 - Phi(2.74) = 1 - 0.9969 = 0.0031.
- +1Decision and conclusion. Since 2.74 > 1.645 (equivalently 0.0031 < 0.05), reject H0. At the 5% level there is sufficient evidence that more than 60% of graduates find a job within three months.
Key terms
- Null hypothesis (H0)
- The status-quo claim being tested; it always contains an equality (e.g. pi = 0.15). We compute the test statistic assuming H0 is true and either reject it or fail to reject it - we never 'accept' it.
- Alternative hypothesis (H1)
- The research claim we look for evidence of (e.g. pi > 0.15 or pi1 != pi2). Its direction decides whether the test is one-tailed or two-tailed.
- Sample proportion (p)
- The observed success rate p = X/n, where X is the number of successes in n trials. Its sampling distribution is approximately normal when n*pi >= 5 and n(1 - pi) >= 5.
- One-sample proportion z-test
- A test of H0: pi = pi0 using Z = (p - pi0) / sqrt(pi0(1 - pi0)/n). The standard error is built from the hypothesised pi0, which distinguishes it from the confidence-interval formula that uses p.
- Pooled proportion (p-bar)
- For a two-sample proportion test under H0: pi1 = pi2, the combined estimate p-bar = (X1 + X2)/(n1 + n2), used to build the standard error of the difference p1 - p2.
- Type I and Type II error
- A Type I error rejects a true H0 (probability alpha, the significance level); a Type II error fails to reject a false H0 (probability beta). Power = 1 - beta. Lowering alpha raises beta.
- p-value
- The probability, if H0 is true, of a test statistic at least as extreme as the one observed. Universal rule: reject H0 when the p-value is less than alpha; otherwise fail to reject.
- prop.test() in R
- The R function that runs proportion tests from success counts and sample sizes. It reports a chi-square statistic equal to z-squared and the same p-value; set correct = FALSE to switch off the continuity correction and match the textbook z-test.
Hypothesis Testing for Business Decisions II FAQ
Why does the proportion test use pi0 in the standard error instead of the sample proportion p?
Because a hypothesis test computes the statistic on the assumption that H0 is true, so it uses the value H0 claims: SE = sqrt(pi0(1 - pi0)/n). A confidence interval makes no such assumption, so its margin of error uses the observed p: sqrt(p(1 - p)/n). Mixing them up is a common lost mark - use pi0 in a test and p in a confidence interval.
How do I choose between one-tailed and two-tailed, and does it change the decision?
Read the alternative hypothesis. Words like 'more than', 'exceeds' or 'less than' point to a one-tailed test (all of alpha in one tail, critical z = 1.645 at 5%). Words like 'differs' or 'is not equal to' point to a two-tailed test (alpha split across both tails, critical z = +/-1.96 at 5%). The choice changes the critical value and the p-value - the same statistic can be significant one-tailed but not two-tailed - so fix the tail before you read the table.
Can AI help me with hypothesis testing for proportions and two-sample tests in ETX5900?
Yes — as a study aid. Sia (AskSia) explains the five-step frame step by step, helps you pick the right test (one proportion, two proportions, or two means), generates fresh practice problems, and checks that the tail and your reject / fail-to-reject decision match the numbers in 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 this as a decision-procedure topic rather than a memorisation one: the exam provides a formula sheet and statistical tables, so marks come from choosing the right test and reading the tables correctly, not from recall. Drill the five-step frame until it is automatic - hypotheses, statistic, critical value / p-value, decision, conclusion in context - and practise fixing the tail from the wording ('more than' one-tailed at 1.645, 'differs' two-tailed at +/-1.96). Be strict about two details that lose easy marks: in a proportion test the standard error uses the hypothesised pi0 (not the sample p), and the two-proportion test pools p-bar under H0. Practise both a reject case and a fail-to-reject case so the direction words ('reject' / 'insufficient evidence', never 'accept H0') come out right, and try each test in R with prop.test() to check your by-hand answer. 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, show the statistic before the number to earn method marks, and confirm the exam date, length and format on Moodle / my.Monash.
Working through Hypothesis Testing for Business Decisions II in ETX5900? Sia is AskSia’s AI Statistics tutor — ask any ETX5900 Hypothesis Testing for Business Decisions II 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.
