SFS Lecture 9 - T-Tests and Confidence Intervals(3)

Lecture Information

• Course: T-tests and Confidence Intervals

• Instructor: Sue Kageler

• Email: Sue.Kageler@uwe.ac.uk

• University: UWE University of the West of England

Learning Outcomes

• By the end of this lecture, students should be able to:

  • Formulate a scientific hypothesis.

  • Apply the Classical Method of Hypothesis Testing:

    • State null and alternate hypotheses.

    • State alpha (the level of significance).

    • Find the rejection region.

  • Carry out a one-sample t-test.

  • Calculate a confidence interval.

Part A: Hypothesis Testing

• Inferential Statistics: Used to make inferences about populations based on sample data.

• Hypotheses:

  • Hypothesis: A proposed explanation made on the basis of limited evidence as a starting point for further investigation.

  • Null Hypothesis (H0): Assumes no change or difference exists.

  • Alternate Hypothesis (H1 or Ha): Indicates a change or difference exists.

Steps in the Classical Method of Hypothesis Testing

1. State null (H0) and alternate hypotheses (H1).

2. Select the test statistic.

3. Select the significance level (α).

4. Construct the decision rule.

5. Carry out calculations.

6. Make a decision.

7. Interpret the decision.

Forms of Hypotheses

• Form I (Two-tailed test):

  • H0: µ = µ0

  • H1: µ ≠ µ0

• Form II (One-tailed test):

  • H0: µ ≥ µ0

  • H1: µ < µ0

• Form III (One-tailed test):

  • H0: µ ≤ µ0

  • H1: µ > µ0

Selecting the Test Statistic

• Choose the statistic used for testing the hypothesis, such as t-test or Chi-Square test.

Selecting the Significance Level (α)

• Choose the α level; represents the probability of rejecting a true null hypothesis (Type I Error).

  • If H0 is true, we:

    • Accept H0: Correct (1 - α).

    • Reject H0: Type I Error (α).

  • If H0 is false, we:

    • Accept H0: Type II Error (β).

    • Reject H0: Correct (1 - β).

Constructing the Decision Rule

• Establishes criteria for rejecting H0 based on critical values:

  • Examples of rejection regions:

    • χ2 > χ2 from table (Chi-Square Test)

    • t > t from table (t test)

Two-tailed and One-tailed Distributions

• Form I (Two-tailed test): H0: µ = µ0, H1: µ ≠ µ0 (Rejection region varies).

• Form II (One-tailed test): H0: µ ≥ µ0, H1: µ < µ0 (Rejection region varies).

• Form III (One-tailed test): H0: µ ≤ µ0, H1: µ > µ0 (Rejection region varies).

Decision Rule: Interpreting p-values

• Compare p value to significance level (α):

  • If p > α: Fail to reject H0.

  • If p < α: Reject H0.

Carrying Out Calculations

• Dependent on the selected test statistic.

Making the Decision

• Determine whether to reject H0 or fail to reject H0.

Interpreting the Decision

• Results must be articulated in relation to the original problem context.

Key Concepts to Master

• Point vs. Interval Estimation:

  • Point Estimation: Single value estimate (e.g., mean).

  • Interval Estimation: Range of values containing the population parameter (e.g., 95% confidence interval).

• Non-Parametric Statistics: Tests with no assumptions about the data distribution (e.g., nominal, ordinal data).

• Independent vs. Related Samples:

  • Independent Samples: Samples do not influence one another.

  • Related Samples: Samples are associated (e.g., paired samples).

Part B: T-tests

• T-tests: Used to compare the means:

  • One sample mean against a specific hypothesized value.

  • The means of two samples to determine if they come from the same population.

T-tests: Assumptions

• Samples must come from a normal distribution.

• Two samples being compared must be independent.

• Variances of samples should follow assumptions to be covered in Level 1.

Hypothesis Test for Population Mean

• Null Hypothesis Form I:

  • H0: µ = µ0

  • H1: µ ≠ µ0

  • Test statistic calculated as:

    ( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} )

Examining Example (1)

• Testing the pH of a buffer solution.

  • Null Hypothesis: No significant difference from stated pH (H0: µ = 3.2).

  • Alternate Hypothesis: Significant difference (H1: µ ≠ 3.2).

  • Test statistic: One-sample t-test.

  • Level of significance: α = 0.05.

Example Continued (2-10)

• Establish rejection region and perform calculations to interpret results.

  • Found significant difference from manufacturer stated pH.

Standard Errors and Test Statistics

• Critical value and degrees of freedom impact rejection criteria for tests.

Part C: Confidence Intervals

• Confidence Intervals: Indicate reliability of sample mean as estimator for population mean.

  • Calculated as:( CI = \bar{x} \pm t_{α/2} \left(\frac{s}{\sqrt{n}}\right) )

Example of Confidence Intervals

• Strawberry yield data used for calculation.

  • 95% CI: Calculate mean and standard deviation to derive intervals.

  • 99% CI: Use increased t-value leading to wider intervals.

Conclusion

• Understanding of hypothesis testing, significance testing, t-tests, and confidence intervals is essential for statistical analysis in research.