One-Sample t-Test

179. One-Sample t-Test

Tests if the mean of a single sample differs from a known or hypothesized population mean.

𝑡 = \frac{\bar{𝑥} - 𝜇_{0}}{\frac{𝑠}{\sqrt{𝑛}}}

Example

[78, 82, 89]

Step 1: State Hypotheses

𝐻_{0} : 𝜇 = 85 cm

𝐻_{1} : 𝜇 \neq 85 cm

Step 2: Summarize Data

𝑥_{1} = 78, 𝑥_{2} = 82, 𝑥_{3} = 89

𝑛 = 3

Step 3: Calculate Sample Mean ( $\bar{𝑥}$ )

\bar{𝑥} = \frac{𝑥_{1} + 𝑥_{2} + 𝑥_{3}}{𝑛} = \frac{78 + 82 + 89}{3} = 83

Step 4: Calculate Sample Standard Deviation:

𝑠 = \sqrt{\frac{\sum_{𝑖 = 1}^{𝑛} {(𝑥_{𝑖} - \bar{𝑥})}^{2}}{𝑛 - 1}}

{(𝑥_{1} - \bar{𝑥})}^{2} = {(78 - 83)}^{2} = {(- 5)}^{2} = 25

{(𝑥_{2} - \bar{𝑥})}^{2} = {(82 - 83)}^{2} = {(- 1)}^{2} = 1

{(𝑥_{3} - \bar{𝑥})}^{2} = {(89 - 83)}^{2} = {(6)}^{2} = 36

SSD = 25 + 1 + 36

𝑠^{2} = \frac{SSD}{𝑛 - 1} = \frac{62}{3 - 1} = \frac{62}{2} = 31

𝑠 = \sqrt{𝑠^{2}} = \sqrt{32} = 5.57

Step 5: Calculate the Test Statistic

𝑡 = \frac{\bar{𝑥} - 𝜇_{0}}{\frac{𝑠}{\sqrt{𝑛}}} = \frac{83 - 85}{\frac{5.57}{\sqrt{3}}} = - \frac{3}{3.22} = - 0.62

Step 6: Determine the Degrees of Freedom

𝑑 𝑓 = 𝑛 - 1 = 15 - 1 = 14

Step 7: Find Critical t-value

For a two-tailed test at a significance level ( $𝛼$ ) of 0.05 and 2 degrees of freedom ( $𝑑 𝑓$ )

4.303

Step 8: Compare the t-Value to the Critical t-Value

If the absolute value of the test statistic is greater than the critical t-value, reject the null hypothesis.
If the absolute value of the test statistic is less than the critical t-value, fail to reject the null hypothesis.

Step 8: Find the p-Value

𝑡 = 4.303 and 𝑑 𝑓 = 3

p-value = 0.58

t_test_one_sample.py

from scipy import stats
rvs = stats.uniform.rvs(size=50)
stats.ttest_1samp(rvs, popmean=0.5)