Unit 5: Sampling Distributions (copy)

Imagine you are trying to learn about a massive group, like every student in your entire state. Here is how to master these concepts step-by-step using simple ideas:

1. Population vs. Sample (Parameter vs. Statistic)

• Population & Parameter: The Population is the entire group. A Parameter is a true, fixed number describing the whole group, like the true population proportion $p$ or true average $\text{μ}$. Memory trick: Parameter goes with Population (P with P).

• Sample & Statistic: A Sample is the smaller group you actually talk to. A Statistic is the number you calculate from your sample, like the sample proportion $ς$ or sample mean $Ł$. Memory trick: Statistic goes with Sample (S with S).

1. Sampling Distribution Imagine taking a random sample of $50$ students, calculating their average height, writing it on a sticky note, and putting them back. If you repeat this thousands of times and plot all those sticky note averages on a graph, that graph is the Sampling Distribution!

2. Bias vs. Variability (The Archery Target)

• Bias: Your shots are systematically off-target (for example, asking only basketball players about average height). Increasing sample size $n$ does NOT fix bias caused by bad sampling.

• Variability: How spread out your shots are. Taking a larger sample size $n$ makes your estimates cluster tighter together, shrinking standard error.

1. Sample Proportions ($ς$)

• Center: $μ_{ς} = p$

• Standard Error: $σ_{ς} = √{ \frac{p(1-p)}{n}}$

• Normal Model Conditions:

  1. Random: Random sample or assignment.

  2. 10% Condition: Sample size $n ≤ 0.10N$ (sample is at most 10% of total population $N$).

  3. Large Counts: Expect at least 10 successes ($np ≥ 10$) and 10 failures ($n(1-p) ≥ 10$).

1. Sample Means ($Ł$) & The Central Limit Theorem (CLT)

• Center: $μ_{Ł} = μ$

• Standard Error: $σ_{Ł} = \frac{σ}{√{n}}$

• Central Limit Theorem (CLT): Even if the original population shape is strange or lopsided, as long as your sample size is large enough ($n ≥ 30$), the sampling distribution of sample means $Ł$ will look like a smooth, symmetrical Normal bell curve!

1. Comparing Two Groups

• For differences ($ς_1 - ς_2$ or $Ł_1 - Ł_2$), the center is simply the difference between parameters ($p_1 - p_2$ or $μ_1 - μ_2$).

• For standard error, always ADD variances under the square root, such as $√{ \frac{σ_1^2}{n_1} + \frac{σ_2^2}{n_2}}$. Never subtract variability!

1. FRQ Probability Checklist To get full credit on test problems:

2. State the distribution name (Normal).

3. State parameters (mean $μ$ and standard error $σ$).

4. State boundary and direction (e.g., $P(Ł > 8.4)$).

5. Show standardized z-score formula: $Z = \frac{ \text{statistic} - \text{parameter}}{ \text{standard error}}$.

6. State the final probability.