The descriptive statistics can be used to determine the inferential statistics.
> c(length(Outcome), mean(Outcome), sd(Outcome))
[1] 8.000000 4.000000 3.116775
The inferential statistics show the key elements to be calculated.
> t.test(Outcome, mu = 7)
One Sample t-test
data: Outcome
t = -2.7225, df = 7, p-value = 0.02966
alternative hypothesis: true mean is not equal to 7
95 percent confidence interval:
1.394311 6.605689
sample estimates:
mean of x
4
Descriptive Statistics: The values of the one-sample statistics are identical to the values that would be provided by the “Descriptives” procedure.
Standard Error of the Mean: The standard error of the mean provides an estimate of how spread out the distribution of all possible random sample means would be.
\[SE_M = \frac{SD}{\sqrt{N}} = \frac{3.117}{\sqrt{8}} = 1.102\]
Mean Difference (Raw Effect): The Mean Difference is the difference between the sample mean and a user-specified test value or population mean.
\[M_{DIFF} = M - \mu = 4.000 − 7.000 = −3.000\]
Statistical Significance: The t statistic is the ratio of the mean difference (raw effect) to the standard error of the mean.
\[t = \frac{M_{DIFF}}{SE_M} = \frac{-3.000}{1.102} = -2.722\]With df = 7, tCRITICAL = 2.365
Because t > tCRITICAL, p < .05
This would be considered a statistically significant finding.
Confidence Interval: For this design, the appropriate confidence interval is around (centered on) the mean difference (raw effect).
\[CI_{DIFF} = M_{DIFF} \pm (t_{CRITICAL} ) (SE_M) = -3.000 \pm (2.365) (1.102) = [ -5.606, -0.394 ]\]Thus, the researcher concludes that the true population mean difference is somewhere between -5.606 and -0.394 (knowing that the estimate could be wrong).
Effect Size: Cohen’s d Statistic provides a standardized effect size for the mean difference (raw effect).
\[d = \frac{M_{DIFF}}{SD} = \frac{-3.000}{3.117} = 0.963\]Given Cohen’s heuristics for interpreting effect sizes, this would be considered a large effect.
For this analysis, a sample mean has been compared to a user-specified test value (or a population mean). Thus, the summary and the inferential statistics focus on that difference.
A one sample t test showed that the difference in Outcome scores between the current sample (N = 8, M = 4.00, SD = 3.12) and the hypothesized value (7.00) was large and statistically significant, t(7) = -2.72, p = .030, 95% CI [-5.61, -.39], d = -0.96.