One Sample T-test

 One Sample T-test

 

Introduction

The One Sample t Test examines whether the mean of a population is statistically different from a known or hypothesized value. The One Sample t Test is a parametric test.

This test is also known as: single sample t test. The variable used in this test is known as: test variable.

Assumption

1. The data are continuous
2. The data are normally distributed
3. The sample is a simple random sample form its population. 

Hypotheses

The null hypothesis (H0) and alternative hypothesis (H1) of the one sample T test can be expressed as:

H0: µ =  µ0  ("the population mean is equal to the hypothesized value")
H1: µ ≠  µ0  ("the population mean is not equal to the hypothesized value")

where µ is the "true" population mean and µ0 is the proposed value of the population mean.

Test Statistic

The test statistic for a One Sample t Test is denoted t, which is calculated using the  following formula: 

μ0= The test value -- the proposed constant for the population mean

x_bar = Sample mean

n = Sample size (i.e., number of observations)

s = Sample standard deviation

s_xbar= Estimated standard error of the mean (s/sqrt(n))

P-value

P-value = Pr(|t_n-1| > |t|) = 2Pr(t_n-1 > |t|)

If p-value is less than 0.05 significant level, then the null hypothesis is true, the population mean is equal to the hypothesized value.

If p-value is greater than 0.05 significant level, then the null hypothesis should be rejected, the population mean is not equal to the hypothesized value.

Relation with Linear Regression

Y = b_0 + e

Then,

H_0 : μ = hypothesized value <=> H_0 : b_0 = hypothesized value

H_1 : μ ≠ hypothesized value <=> H_1 : b_0 ≠ hypothesized value

R code

(from STA302 assignment)

Here is the example R code of one sample t-test about the relation between forearm length and height.

 

Reference

https://www.slideshare.net/ebogue/onesample-hypothesis-tests

https://libguides.library.kent.edu/SPSS/OneSampletTest