A common workflow in psychology statistics is testing whether a sample mean differs from a benchmark (a norm score, clinical cutoff, or historical average) when the population standard deviation is unknown. This is handled by a one-sample \(t\)-test and the \(t\) confidence interval for the mean.
Anxiety scores are measured after an intervention. A benchmark value of 50 represents the norm for the test. A sample of \(n=16\) participants yields \(\bar{x}=42\) and sample standard deviation \(s=8\). The research question is whether the intervention produces a mean below the benchmark (directional claim).
1) State hypotheses and significance level
- Null hypothesis: \(H_0:\mu=50\)
- Alternative hypothesis (left-tailed): \(H_a:\mu<50\)
- Significance level: \(\alpha=0.05\)
2) Compute the test statistic
The one-sample \(t\) statistic (σ unknown) is \[ t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}. \]
| Quantity | Symbol | Value |
|---|---|---|
| Sample size | \(n\) | 16 |
| Sample mean | \(\bar{x}\) | 42 |
| Sample standard deviation | \(s\) | 8 |
| Benchmark (hypothesized mean) | \(\mu_0\) | 50 |
Standard error: \[ \text{SE}=\frac{s}{\sqrt{n}}=\frac{8}{\sqrt{16}}=\frac{8}{4}=2. \]
Test statistic: \[ t=\frac{42-50}{2}=\frac{-8}{2}=-4. \]
3) Degrees of freedom and the p-value
Degrees of freedom: \(df=n-1=16-1=15\).
For a left-tailed test, the p-value is \[ p=P\!\left(T_{15}\le -4\right). \]
Since \(p \lt 0.05\), the null hypothesis is rejected. The data provide strong evidence (under the model assumptions) that the population mean anxiety score is below 50.
4) Equivalent critical-value decision (optional check)
For a left-tailed test at \(\alpha=0.05\) with \(df=15\), the critical value is approximately \[ t_{\text{crit}}=-t_{0.95,15}\approx -1.753. \]
Because \(t=-4 \lt -1.753\), the test statistic falls in the rejection region, matching the p-value conclusion.
5) 95% confidence interval for the population mean
A two-sided 95% confidence interval for \(\mu\) is \[ \bar{x}\pm t^\*\cdot \frac{s}{\sqrt{n}}, \quad t^\*=t_{0.975,15}\approx 2.131. \]
\[ 42 \pm 2.131 \times 2 = 42 \pm 4.262 \quad \Rightarrow \quad (37.738,\ 46.262). \]
The entire interval lies below 50, consistent with the hypothesis test result.
6) Effect size commonly reported in psychology statistics
A standard one-sample effect size is Cohen’s \(d\): \[ d=\frac{\bar{x}-\mu_0}{s}=\frac{42-50}{8}=\frac{-8}{8}=-1.0. \]
The sign indicates direction (below the benchmark). The magnitude \(|d|=1.0\) indicates a large standardized difference under common conventions, although practical interpretation should be tied to the measurement scale and context.
Visualization: t test decision on a curve
Assumptions and reporting notes
- Independence: observations represent independent participants or independent measurements.
- Approximate normality: the population distribution is approximately normal, or the sample size is adequate for robustness; strong skew/outliers can distort the result.
- Scale interpretation: many psychology measures are treated as interval for mean-based inference; justification should match the instrument and study design.
- Report together: \(t(15)=-4.00\), \(p\approx 0.0006\) (one-tailed), 95% CI \((37.738, 46.262)\), and \(d=-1.0\).