Unit 4: Hypothesis testing and inferential statistics
Business Research Methods notes · PTU syllabus (MCOP203-18)
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Unit summary
Hypothesis testing lets researchers decide whether patterns in data are real. This unit covers formulating and testing hypotheses, Type I and Type II errors, t, Z, F and chi-square tests, goodness of fit, ANOVA, association of attributes, partial and multiple correlation, multiple regression, and testing regression assumptions — multicollinearity, heteroscedasticity and autocorrelation.
After this unit you can
- Formulate hypotheses and explain the testing procedure and errors
- Apply Z, t, F, chi-square tests and ANOVA
- Measure association of attributes and partial and multiple correlation
- Explain multiple regression and the testing of its assumptions
PTU syllabus topics
- Hypothesis formulation and testing procedure
- errors in hypothesis testing
- t-test/Z-test/F-test/chi-square test/goodness of fit/ANOVA
- techniques of association of attributes
- partial and multiple correlation
- multiple regression
- testing assumptions of regression (multicollinearity, heteroscedasticity, autocorrelation)
Z-test
Mean with large sample (n ≥ 30) or known σ
Interval or ratio
t-test
Mean with small sample
Interval or ratio
F-test / ANOVA
Comparing variances or 3+ means
Interval or ratio
Chi-square
Association or goodness of fit
Counts (nominal)
Topic 1
Hypothesis testing procedure and errors
- 1
State null (H0) and alternative (H1) hypotheses
- 2
Choose level of significance (α — 5% or 1%)
- 3
Select the test statistic
Z, t, F, χ²
- 4
Determine the critical region
One-tailed or two-tailed
- 5
Compute the test statistic from sample data
- 6
Decide
Reject H0 if statistic falls in the critical region (or p-value < α)
- 7
Interpret in business terms
Meaning
Rejecting a true H0
Accepting a false H0
Example
Concluding a new drug works when it does not
Missing a drug that actually works
Controlled by
Level of significance
Sample size and power (1 − β)
Topic 2
Parametric tests: Z, t, F and ANOVA
| Test | Used for | Statistic |
|---|---|---|
| Z-test | Mean or proportion, large samples (n ≥ 30) or σ known | Z = (x̄ − μ) ÷ (σ/√n) |
| t-test (one sample) | Mean, small sample, σ unknown | t = (x̄ − μ) ÷ (s/√n), df = n − 1 |
| t-test (two independent samples) | Difference of two means | Pooled variance, df = n1 + n2 − 2 |
| Paired t-test | Before–after on the same units | t = d̄ ÷ (sd/√n) |
| F-test | Equality of two variances | F = s1² ÷ s2² (larger on top) |
| ANOVA | Equality of three or more means | F = MSB ÷ MSW |
Example
A sample of 25 bulbs has mean life 1,520 hours, s = 100; claim μ = 1,500. t = 20 ÷ 20 = 1.0 < 2.064 (5%, df 24) → do not reject the claim.
- 1Total variation (SST)
- 2Between-group variation (SSB)
- 3Within-group variation (SSW)
- 4Mean squares
MSB = SSB ÷ (k − 1); MSW = SSW ÷ (N − k)
- 5F = MSB ÷ MSW compared with table F
- Two-way ANOVA tests two factors (e.g., salesperson and region) and their interaction.
Topic 3
Chi-square test and association of attributes
- Chi-square (χ²) = Σ (O − E)² ÷ E — non-parametric.
- Uses: goodness of fit (does data follow a distribution? df = k − 1 − parameters estimated), test of independence in contingency tables (df = (r − 1)(c − 1)), test of population variance.
Example
Survey of 200: preference for brand X by gender. If the computed χ² = 7.2 with df = 1 and the 5% table value is 3.84, gender and brand preference are associated.
- Association of attributes: Yule's coefficient of association Q = (AD − BC) ÷ (AD + BC) — ranges from −1 to +1; coefficient of colligation; contingency coefficient.
Topic 4
Partial and multiple correlation; multiple regression
Correlation measures the degree of relationship between two variables.
Positive
Both move in the same direction
Advertising and sales
Negative
Move in opposite directions
Price and demand
Simple vs multiple
Two variables vs three or more
Yield on rainfall vs on rainfall and fertiliser
Partial
Two variables, holding others constant
Yield and rainfall with temperature fixed
Linear vs non-linear
Constant vs changing ratio of change
Straight line vs curve
A scatter diagram plots pairs of values; the pattern of dots shows the direction and strength of the relationship.
- Simple correlation: two variables. Multiple correlation: one variable with two or more others jointly (R1.23). Partial correlation: two variables with the effect of others held constant (r12.3).
Partial correlation
r12.3 = (r12 − r13 r23) ÷ √[(1 − r13²)(1 − r23²)]
Multiple correlation
R1.23 = √[(r12² + r13² − 2 r12 r13 r23) ÷ (1 − r23²)]
Multiple regression
- Model: Y = b0 + b1X1 + b2X2 + … + bkXk + e — estimated by ordinary least squares (OLS).
- Interpretation: bi is the change in Y for a unit change in Xi, holding other variables constant; R² — proportion of variation explained; adjusted R² penalises extra variables; F-test for overall significance; t-tests for individual coefficients.
Multicollinearity
High correlation among independent variables — detect by VIF > 10, correlation matrix; remedy — drop/combine variables
Heteroscedasticity
Non-constant error variance — detect by residual plots, Breusch–Pagan, White test; remedy — robust standard errors, transformations
Autocorrelation
Errors correlated over time — detect by Durbin–Watson (≈ 2 means none); remedy — add lags, GLS
Normality of errors
Jarque–Bera test
Linearity and correct specification
Exam tip
In interpreting software output, read in order: F-test (model), R²/adjusted R², each coefficient's sign and t-value, then diagnostics (VIF, DW).
Key terms
- Null hypothesis
- Statement of no effect or no difference
- Level of significance
- Probability of a Type I error
- ANOVA
- Test for equality of several means using variance components
- Chi-square test
- Test comparing observed and expected frequencies
- Multicollinearity
- High correlation among explanatory variables
Quick revision
- Hypothesis steps; Type I (α) and Type II (β) errors.
- Z (large n), t (small n), F (variances), ANOVA (several means).
- χ²: goodness of fit and independence; Yule's Q.
- Multiple regression: R², adjusted R², F and t tests.
- Diagnostics: VIF, Breusch–Pagan, Durbin–Watson.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Distinguish null and alternative hypotheses.
- Q2.What is a Type I error?
- Q3.When is a t-test used?
- Q4.What is ANOVA?
- Q5.State Yule's coefficient of association.
- Q6.What is heteroscedasticity?
Long-answer questions
- Q1.Explain the procedure of hypothesis testing and types of errors.
- Q2.Explain t, Z and F tests with examples.
- Q3.Explain ANOVA and the chi-square test with applications.
- Q4.Explain multiple regression and the testing of its assumptions.
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