one sample t test

Overview of Testing Structure

• Conducting about one significant testing each day for nine tests total.

• The procedure involves five steps of null hypothesis significance testing.

• Variations will mainly involve the type of test and associated sampling distributions.

Review Question: One Sample Z Test

• What research question does a one sample z test answer?

• It assesses the difference between a sample mean and a hypothesized population mean, assuming the population standard deviation is known.

Critical Values in Testing

• The critical value of 1.96 serves as a threshold for reject/retain decisions for null hypothesis.

• This value indicates where the rare zone starts in the z distribution.

Limitations of Z Tests

• Z tests are often not practical for psychologists since obtaining the population standard deviation is challenging.

• Instead, we use sample data to estimate the population parameter, leading to some degree of uncertainty.

• Smaller samples result in greater uncertainty and variability in estimation.

T Distributions

• T distributions are a family of distributions that vary based on degrees of freedom which is related to sample size (n).

• The shape of the t distribution varies; smaller sample sizes produce heavier tails due to increased variability.

• As sample size increases, the t distribution approaches the normal distribution.

Degrees of Freedom

• Defined as the number of scores that can vary in estimating a parameter from the sample: ( df = n - 1 )

• Example: For three numbers with a mean of five – if two numbers are known, the third is determined, leaving two degrees of freedom.

Normal vs. T Distribution

• T values akin to critical z values vary with sample sizes.

• Z critical value is 1.96 while t values increase with smaller sample sizes (e.g., ( df = 10 ) gives ( t = 2.088 )).

Research Example: Meditation on Stress Levels

• A study measuring stress before and after meditation over six months

• Population mean stress level is 6 vs. the meditation group mean of 5.

• Examination of whether the difference is statistically significant or due to sampling error.

Choosing the Appropriate Test

• For comparing a sample mean to a population mean with an unknown population standard deviation, a one sample t test is utilized.

Assumptions of One Sample T Test

• Random sampling from the population (robust if violated).

• Independence of observations (not robust; violations lead to unreliable conclusions).

• Normality of the dependent variable within the population (robust if sample size is large).

Statistical Hypotheses

• Null hypothesis (H0): Population mean of meditators is 6 (no difference).

• Alternative hypothesis (H1): Mean is not 6 (there is a difference).

Decision Rule

• Set significance level (( b1 = 0.05 )).

• Find critical t value based on degrees of freedom (n - 1).

• Example: For ( n = 8 ), degrees of freedom is 7, yielding a critical t value of approximately 2.365.

• If the calculated t statistic is beyond the critical t, reject the null hypothesis.

Test Statistic Calculation

• Formula: ( t = \frac{m - \mu}{s_m} ) where:

• ( m ) = sample mean,

• ( \mu ) = population mean,

• ( s_m ) = estimated standard error of the mean.

• Results showed a calculated t of -1.37, indicating a common zone under the null hypothesis.

P-Value and Confidence Intervals

• A p-value greater than 0.05 indicates retaining the null hypothesis.

• Calculated Confidence Interval: ( 95\% \text{ CI } = (3.27, 6.73) ) interpreting that we are 95% confident the true mean lies in this range.

• Significant because the population mean of 6 falls within this interval.

Effect Sizes

• Effect sizes quantify the strength of a relationship, e.g., using Cohen's d for t tests.

• P-value alone does not measure effect strength; larger sample sizes can yield significant results for smaller effects.