Assignment III - PART A - Probable Answers

PART A (Add your own examples and formula in exam)

Here are some acceptable definitions/concepts for the questions:

1. Bias in Estimation: In statistics, bias refers to the systematic tendency of an estimator to consistently overestimate or underestimate the true value of the parameter it's trying to represent. Imagine you're trying to estimate the average height of people in your city. If your sample only includes people from a basketball team, the average height will likely be biased upwards because it doesn't represent the entire population.

2. Mean Squared Error (MSE): This term measures the average squared difference between the estimated values and the actual values. A lower MSE indicates that the estimator, on average, produces predictions closer to the true values. Think of it like an average squared distance between your "guesses" and the bullseye in archery.

3. Two-way ANOVA (Analysis of Variance): This statistical test helps us understand how two categorical factors simultaneously affect a continuous outcome variable. It analyzes the variance (spread) in the data to determine if the differences in means between groups are due to chance or if there's a statistically significant effect from one or both factors, or their interaction. Imagine studying the effect of fertilizer type (A) and watering frequency (B) on plant growth (outcome). A two-way ANOVA would tell you if either factor alone, or both together, significantly influence plant growth.

4. Confidence Intervals: These are ranges of values that are likely to contain the true population parameter with a specific level of confidence (often 95% or 99%). It's like estimating a target's location on a dartboard by throwing multiple darts. The confidence interval is the area where you're reasonably certain the bullseye lies, based on the spread of your throws.

5. Bias vs. Precision:

   • Bias: As mentioned earlier, bias is the systematic error that causes an estimator to consistently deviate from the true value. It reflects how accurate your estimates are on average.
   • Precision: This refers to how close repeated measurements are to each other. It indicates the level of random variation in your estimates. Think of it like the tightness of your dart throws around the target. High precision means your throws are clustered together, even if they're not necessarily hitting the bullseye (due to bias).

6. Parameter: A parameter is a numerical characteristic that describes a population. It's a fixed but unknown value we're trying to estimate. For example, the population mean (average height of all people in your city) is a parameter.

7. One-way ANOVA: This is a statistical test used to compare the means of several groups (more than two) to see if there's a statistically significant difference between their averages. It assumes there's only one factor influencing the outcome variable. Imagine comparing the average exam scores of students taught by different teachers. A one-way ANOVA would tell you if there's a significant difference in teaching effectiveness based on average scores.

8. Sample Size: This refers to the number of observations or data points included in your statistical analysis. A larger sample size generally leads to more reliable estimates and more accurate conclusions from your analysis. It's like having more dart throws; the more throws you have, the better you can estimate the target's location.