AGRI 2400: Experimental Methods in Agricultural and Food Sciences - Key Concepts and Statistical Tests

AGRI 2400 Experimental Methods Notes

Standard Error of the Mean (SEM)

• Definition: SEM measures the accuracy with which a sample represents a population. It calculates the range where sample means will occur at a certain confidence level (e.g., 95%).

• The formula for SEM:

  • Represents the range within which 95% of sample means will lie based on the normal distribution.

Analysis 1: The Z Test

• Purpose: Used to determine if a sample is drawn from a population with a known mean and standard deviation.

• Usage: Applicable for comparisons to historical data or established standards (e.g., physiology measures).

• Assumptions:

  • Population standard deviation (σ) is known.

  • Samples are independent and randomly selected from their populations.

  • Data is measured on an interval or ratio scale.

  • Data is normally distributed or has a large enough sample size to assume normality.

  • No outliers in the data.

Z Test Calculation Steps

1. Calculate SEM:

2. Calculate 95% confidence interval:

3. Check if sample mean falls within this interval.

Interpreting z-test Results

• If is within the confidence interval:

  • Null hypothesis (H0) is retained, suggesting no significant difference from the population mean (p > 0.05).

• If is outside the confidence interval:

  • Null hypothesis is rejected, indicating a significant difference (p < 0.05).

Example Z Test

• Context: White blood cell count in dairy cows.

  • Known total mean: 7500 white blood cells/ml, SD: 1250.

  • Sample mean from 10 previously ill cows: (\bar{X} = 7310.5).

Calculating:

1. Calculate SEM:

2. Calculate confidence interval:

3. Determine if sample mean is within range: 7310.5 is within the interval.

• Purpose: Compares a sample mean against a population mean when population parameters are unknown.

• Assumptions:

  • Independent random samples.

  • Data on ratio or interval scale.

  • No outliers; approximately normal distribution or sufficient sample size.

One Sample t-Test Steps

1. Calculate SEM:
   Calculate t statistic:

2. Compare t with critical values from t-table.

Reporting Results from t-Test

• Report:

  • Mean, SD, sample size,

  • t statistic,

  • Degrees of freedom,

  • p-value for your conclusion.

• Example Statement: "Mean water content was not 18% (t = 2.49, df = 8, p < 0.05), indicating harvest should not occur."

Analysis 3: Paired Sample t-Test

• Purpose: Compare means from two related samples (e.g., pre-and post-tests).

• Assumptions: Same as one sample t-test.

Reporting Paired Sample t-Test Results

• Example statement: "No significant difference in time taken to consume maggots (t = 0.22, df = 11, p > 0.05) suggests diet preference may require more encounters."

Analysis 4: Independent Samples t-Test

• Purpose: Compare means between two independent groups.

• Assumptions:

  • Homogeneity of variances (equal variances).

  • Independent samples.

  • Normal distribution of dependent variable.

Reporting Independent Samples t-Test Results

• Example: "Mean weight gain was significantly higher on the new diet (26.62 lbs) compared to control (21.65 lbs) (Welch’s t-test, t(48) = -2.87, p = 0.006)."

Important Considerations

• Use statistical software for complex calculations.

• Always check assumptions before selecting the appropriate test.

• Be cautious with one-tailed tests: specify hypotheses beforehand to prevent misinterpretation of results.

Next Steps

• Prepare for ANOVA: Single factor CRD analyses in the next lecture.