One-Sample t-Test Task
A One-Sample t-Test Task is a statistical hypothesis testing task whose test statistic is a one-sample t-test.
- AKA: Single-Sample t-Testing.
- Context:
- Task Input:
- Input Data : [math]\displaystyle{ \{x_1,x_2,\cdots,x_n\} }[/math], sample data.
- Input Parameters:
- [math]\displaystyle{ \mu_0 }[/math], a hypothesized population mean value.
- [math]\displaystyle{ \alpha_0 }[/math], a significance level value or a confidence level (a percentage).
- (optional) [math]\displaystyle{ \sigma^2 }[/math], a population variance value.
- output:
- one-sample t-test statistic value,
- P-value or Region of Acceptance
- Region of Rejection (optional)
- Decision Errors (optional).
- Task Requirements
- Hypotheses Statement: a null hypothesis and an alternative hypothesis according to one-tailed one-sample t-test or two-tailed one-sample t-test
- Test Statistic computation: This require the calculation an one-sample t-test statistic from a sample mean and sample standard deviation.
- P-value and/or Region of acceptance computation: these require a t-distribution calculator or t-table.
- Decision Rule: Null hypothesis is rejected if P-value is less than [math]\displaystyle{ \alpha_0 }[/math] or if the t-test statistic value follows outside region of acceptance.
- It can be solved by a One-Sample t-Test System (that implements a one-sample t-test algorithm).
- Task Input:
- Example(s):
- Let's consider the dataset corresponding Verbal Intelligent Quotient (VIQ) measure retrieved from http://www.scipy-lectures.org/_downloads/brain_size.csv and the One-Sample t-Test System based on the scipy statistical function
stats.ttest_1samp()
[1] then:- Task input: sample data =
data['VIQ']
, [math]\displaystyle{ \mu_0=0 }[/math] - Task output: t-statistic = 30.08809997084934, p-value=1.3289196468727784e-28
- For [math]\displaystyle{ \alpha_0=0.001,0.01,0.025, 0.05 }[/math] the null hypothesis is rejected as p-value is less than significance level (because the p-value is too small, we can concluded the population mean is not 0).
- Task input: sample data =
- …
- Let's consider the dataset corresponding Verbal Intelligent Quotient (VIQ) measure retrieved from http://www.scipy-lectures.org/_downloads/brain_size.csv and the One-Sample t-Test System based on the scipy statistical function
- Counter-Example(s):
- See: One-Sample t-Test System, Statistical Significance, Sample Average, Sample Variance.
References
2017
- (Stat Trek, 2017) ⇒ http://stattrek.com/statistics/dictionary.aspx?definition=One-sample%20t-test
- A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value.
(...) The test statistic is a t statistic (t) defined by the following equation.
- A one-sample t-test is used to test whether a population mean is significantly different from some hypothesized value.
- [math]\displaystyle{ t = \frac{(x - M )}{s \sqrt{n}} }[/math]
- where [math]\displaystyle{ x }[/math] is the observed sample mean, [math]\displaystyle{ M }[/math] is the hypothesized population mean (from the null hypothesis), and [math]\displaystyle{ s }[/math] is the standard deviation of the sample.
2011
- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-test#One-sample_t-test
- In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic
- [math]\displaystyle{ t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}} }[/math]
- where [math]\displaystyle{ \overline{x} }[/math] is the sample mean, s is the sample standard deviation of the sample and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means, [math]\displaystyle{ \overline {x} }[/math], is assumed to be normal. By the central limit theorem, if the sampling of the parent population is independent then the sample means will be approximately normal.[1] (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)
- ↑ George Box, William Hunter, and J. Stuart Hunter, Statistics for Experimenters, ISBN 978-0471093152, pp. 66–67.