One-sample t-Test and Confidence Interval with Dot Chart in Small Samples
Prepared by Douglas Woolford and Ian McLeod
The dot chart represents a random sample of size = 9 from a normal distribution, shown in blue, with mean μ0 + δ and standard deviation σ. This is the true distribution. This Demonstration focuses on the null hypothesis, H0: μ = μ0. For hypothesis testing, it is assumed that the sample is from a normal population with unknown mean and unknown variance. The hypothetical distribution, N(μ0,σ), is shown in green. When δ = 0, the null distribution and true distribution are the same, but otherwise they are different.
The gray curve shows the sampling distribution. The (1 - α)% two-sided confidence interval based on the random sample is shown below with α = 0.05. The confidence interval is colored red, to indicate that the hypothesized mean μ0 is not included in the interval; otherwise the color is blue. So when δ = 0, this confidence interval will be blue about 100(1 - α)% of the time and for larger |δ| this happens less frequently.
The observed two-sided p-value is shown at the top.
By using the "random seed" slider, we can see that the p-values are random and depend on the random sample. The location and width of the confidence interval as well as the sampling distribution also vary and depend on the random sample. Both of these widths tend to decrease as the sample size increases but there is considerable variation due to random sampling. Similarly, when α is increased or decreased the width of the confidence interval tends to increase or decrease.
The sample mean is represented by the thin magenta line extending from the midpoint of the confidence interval to the center value of the sampling density.
The width of the confidence tends to increase if α is decreased, although due to the small samples there is a large variation simply due to random sampling. This is noticeable by changing the random seed.
The confidence interval also tends to increase if the sample size n is decreased as does the width of the sampling distribution. Due to the small samples, there are large variations in the width of the confidence interval and sampling distribution.