module 3 stat part 1

One Sample T Test Overview

• The One Sample T Test is used to compare a single mean to an external population value.

• It is primarily applied to continuous data, where mean values can be calculated.

• This test is considered a parametric test, which assumes a specific distributional form (the t distribution).

• Different versions of the t test exist depending on hypothesis and study designs (forthcoming topics).

Key Concepts

• Continuous Data: Data that can take any value within a given range, allowing for the calculation of means.

• T Test: A statistical method to determine if there is a significant difference between the means of two groups, based on assumptions about the data's distribution.

• T Distribution: A distribution used in the context of the t test, especially when dealing with small sample sizes.

Test Hypothesis

• The one sample t test evaluates if the mean of a sample is equal to a specific, fixed value.

  • Null Hypothesis (H0): The sample mean is equal to the fixed value (e.g., lung peak airflow = 300 ml/sec).

  • Alternative Hypothesis (H1): The sample mean is not equal to the fixed value.

Calculation of T Statistic

• The formula for the t statistic is:

  • t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

  • Example calculation: For a sample mean of 288, hypothesized mean of 300, sample standard deviation of 18, and a sample size of 28:

    • t = (288 - 300) / (18 / sqrt(28))

    • Result: t ≈ -3.52

Interpretation of T Value

• A smaller t value indicates that the sample mean is close to the hypothesized value.

• A larger t value (in absolute terms) suggests a significant difference between the sample and hypothesized means.

• Here, |-3.52| suggests a significant difference from 300 ml/sec.

Standardization

• The t value represents the number of standard errors the sample mean is from the hypothesized mean (300 ml/sec).

• A t value of -3.52 indicates that the hypothesized mean is 3.52 standard errors away from the sample mean.

P Value

• The significance of the t statistic is evaluated via the p value.

• In this case, a p value of 0.0016 indicates a low probability of observing this difference if the null hypothesis is true.

• This suggests the possibility of rejecting the null hypothesis, leading to the conclusion that the peak airflow does not equal 300 ml/sec.

Summary of One Sample T Test

• The one sample t test compares a mean to a specific reference value (hypothesized or external).

• Detailed understanding of calculating the t statistic and interpreting its implications is crucial.

• Practical implementation can be facilitated using statistical software such as R Commander for analysis in exercises.

Upcoming Topics

• Next session will dive deeper into the Student's t distribution.

• Discussion on degrees of freedom and its effect on the t distribution shape will be provided in subsequent content.