A single sample t-test is a statistical hypothesis test used to determine if the mean of a population is equal to a specified value. It is a widely used statistical tool in various fields, such as psychology, medicine, and economics. The single sample t-test assumes the normality of the data and requires only one sample from the population. The test statistic is calculated as the difference between the sample mean and the hypothesized population mean, divided by the sample standard deviation. The resulting t-value is compared to a critical value from the t-distribution with the appropriate degrees of freedom to determine statistical significance.
What is a Single Sample t-Test?
A single sample t-test is a statistical hypothesis test used to compare the mean of a single sample to a known population mean or to a hypothesized value. It is used when the population standard deviation is known and the sample size is small (typically less than 30).
Steps to Perform a Single Sample t-Test:
-
State the null and alternative hypotheses:
- Null hypothesis (H0): The mean of the sample is equal to the population mean (μ0).
- Alternative hypothesis (Ha): The mean of the sample is not equal to the population mean (μ ≠ μ0).
-
Calculate the test statistic (t):
- t = (x̄ – μ0) / (s / √n)
- where:
- x̄ is the sample mean
- μ0 is the known population mean or hypothesized value
- s is the sample standard deviation
- n is the sample size
-
Determine the degrees of freedom (df):
- df = n – 1
-
Find the critical value (tα/2):
- Using a t-table with df and a given significance level (α).
-
Make a decision:
- If the absolute value of the test statistic (|t|) is greater than the critical value (|t| > |tα/2|), reject the null hypothesis and conclude that the sample mean is different from the population mean.
- Otherwise, fail to reject the null hypothesis and conclude that there is not enough evidence to say that the sample mean is different from the population mean.
Example:
Null hypothesis: The average height of adult males in a certain population is 175 cm.
Sample: A random sample of 25 adult males has a mean height of 178 cm and a standard deviation of 5 cm.
Test statistic:
t = (178 – 175) / (5 / √25) = 3
Degrees of freedom:
df = 25 – 1 = 24
Critical value:
t0.025,24 = 2.064
Decision:
|t| (3) > |t0.025,24| (2.064). Therefore, we reject the null hypothesis and conclude that the average height of adult males in this sample is different from the hypothesized population mean of 175 cm.
Question 1:
What is the purpose of a single sample t-test?
Answer:
A single sample t-test is a statistical hypothesis test used to determine whether the mean of a single population is significantly different from a known value.
Question 2:
What are the assumptions of a single sample t-test?
Answer:
The assumptions of a single sample t-test are that the population is normally distributed, the sample is random, and the sample size is large enough (typically at least 30).
Question 3:
What is the null hypothesis in a single sample t-test?
Answer:
The null hypothesis in a single sample t-test is that the population mean is equal to the hypothesized value.
Well, there you have it, folks! That’s a quick and dirty rundown of what a single sample t-test is all about. Hope it cleared things up a bit. If you’re still scratching your head, don’t worry, you’re not alone. Statistics can be a bit mind-boggling sometimes. Just remember, it’s all about making sense of data and drawing meaningful conclusions. Thanks for reading, and be sure to swing by again later for more statistical adventures!