MEDD8815 Chap.5 Inference for Means with t Procedures
Inference for Means with t Procedures
From z to t
Session 5 removes the unrealistic assumption that the population standard deviation is known. Replacing it with the sample standard deviation changes the reference distribution from the standard Normal curve to a t distribution with n minus 1 degrees of freedom.
The t curves are symmetric like the Normal curve but have heavier tails, most noticeably for small samples, and Table D gives their critical values by degrees of freedom and tail area.
Three designs, three procedures
The chapter organises the session around one decision. A single sample compared with a fixed value uses the one-sample t procedure.
Two measurements on the same individuals, or on matched pairs, are reduced to their differences and analysed with the one-sample procedure on those differences. Two separate groups use the two-sample t procedure, whose standard error adds the two sample variances divided by their sample sizes.
Each is worked from AskSia-authored data with Table D bracketing of the P-value.
Robustness, degrees of freedom and pooling
The t procedures are robust to moderate non-Normality, particularly in larger samples, but not to outliers, and random sampling matters more than Normality.
For two samples the slides give a conservative choice of degrees of freedom, the smaller of the two sample sizes minus one, alongside the software approximation.
The pooled procedure assumes equal population standard deviations; the slides call the unpooled version safer, while the session homework instructs students to assume equal variances, and the chapter shows how to follow each instruction.
The SPSS routines
The session handout covers the One-Sample, Paired-Samples and Independent-Samples T Test dialogs.
Two details decide many answers: SPSS reports two-sided P-values, which must be halved for a one-sided alternative in the predicted direction, and the one-sample interval SPSS prints is for the difference from the test value, so the test value must be added back. The handout's shift trick for testing a non-zero paired difference is also explained.
What this chapter covers
- 01
Why an unknown sigma leads to t distributions
- 02
Reading Table D for critical values and P-values
- 03
One-sample t confidence intervals and tests
- 04
Matched pairs analysed through differences
- 05
Robustness and when t procedures are safe
- 06
Two-sample t statistic and conservative degrees of freedom
- 07
Pooled two-sample procedures
- 08
Running and reading the t tests in SPSS
Worked example · free
Paired or independent: the same numbers, two answers
- 2Compute the differences and their mean, standard deviation and standard error.
- 2Compute t, state the degrees of freedom and bracket the one-sided P-value.
- 2Explain the consequence of using a two-sample procedure instead.
Key terms
- t Distribution
- A symmetric, single-peaked distribution with heavier tails than the standard Normal, indexed by its degrees of freedom and used when sigma is estimated by s.
- Degrees of Freedom
- The number that identifies which t, chi-square or F distribution applies; for a one-sample t procedure it is n minus 1.
- Standard Error
- The estimated standard deviation of a statistic, such as s divided by the square root of n for a sample mean.
- Matched Pairs Design
- A design in which each individual is measured twice, or individuals are paired, so that the analysis is carried out on the within-pair differences.
- Robust Procedure
- A procedure whose confidence level or P-value changes little when its conditions, such as Normality, are not exactly met.
- Pooled Standard Deviation
- A single estimate of a common population standard deviation formed by combining the sample variances of two or more groups, weighted by their degrees of freedom.
Inference for Means with t Procedures FAQ
How do I halve an SPSS P-value correctly?
SPSS prints a two-sided Sig. value. If your alternative is one-sided and the sample result lies in the direction it predicts, divide Sig. by 2. If the sample result lies in the opposite direction, the one-sided P-value is one minus half of Sig., and the data give no support to the alternative.
Should I use the pooled or the unpooled two-sample test?
The slides describe the unpooled version as the safer default because equal population variances are hard to verify, while the Session 5 homework tells you to assume equal variances. Follow the instruction attached to the task in front of you, and if the two SPSS rows disagree, report both and say which you relied on.
What conservative degrees of freedom should I use by hand?
Use the smaller of the two sample sizes minus one. It produces a slightly larger P-value than the software approximation, so a conclusion reached with it is safe. SPSS and other software compute a fractional value that is usually larger and gives a slightly smaller P-value.
Assessment move
Write the three standard errors, one sample, paired and two independent samples, on a card with the degrees of freedom beside each. Before every problem in this session, write which of the three you are using and why before touching the numbers. Most lost marks here come from a correct calculation on the wrong design.
Working through Inference for Means with t Procedures in MEDD8815? Sia is AskSia’s AI Statistics tutor — ask any MEDD8815 Inference for Means with t Procedures question and get a clear, step-by-step explanation grounded in how MEDD8815 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
