APS Section 7.3: Sample Means Study Guide

Learning Targets for Section 7.3: Sample Means

• Calculate Mean and Standard Deviation: Be able to calculate ̄{x} and interpret the standard deviation of the sampling distribution of a sample mean.
• Examine Shape Influences: Explain how the shape of the sampling distribution of ̄{x} is affected by the population distribution shape and the sample size.
• Difference in Sample Means: Calculate the mean and standard deviation of the sampling distribution of a difference in sample means ̄{x}_1 - ̄{x}_2 and interpret that standard deviation.
• Normality Assessment: Determine if the sampling distribution of ̄{x}_1 - ̄{x}_2 is approximately Normal.
• Probability Calculations: If appropriate, use a Normal distribution to calculate probabilities involving ̄{x} or ̄{x}_1 - ̄{x}_2.

Introduction to Quantitative Variables and Sample Statistics

• Categorical vs. Quantitative Variables:   - Sample proportions ($\hat{p}$) typically arise when investigating categorical variables (e.g., "What proportion of adults watched a specific show?").   - Quantitative variables (e.g., household income, blood pressure, lifetime of car brake pads) require different statistics such as the median, mean, or standard deviation.
• The Sample Mean: The sample mean ̄{x} is the most common statistic computed from quantitative data.

The Sampling Distribution of ̄{x}

• Definition: The sampling distribution of the sample mean ̄{x} describes the distribution of values taken by the sample mean ̄{x} in all possible samples of the same size from the same population.
• Activity Case Study: "Penny for Your Thoughts":   - Activity conducted by Mrs. Gallas’s class using a population of pennies.   - Students produced dotplots for samples of size $n = 5$ and samples of size $n = 20$.
• Comparison of Sampling Distributions from the Activity:   - Shape: The distribution is slightly skewed to the left when $n = 5$, but becomes roughly symmetric when $n = 20$.   - Center: The distribution is centered at approximately the year $2002$ for both sample sizes (̑{x} \approx 2002).   - Variability: The distribution of ̄{x} is about half as variable when using samples of size $n = 20$ (σ_{̄{x}} \approx 2.6) compared to samples of size $n = 5$ (σ_{̄{x}} \approx 5.2).

Mean and Standard Deviation of the Sampling Distribution of ̄{x}

• General Principle: Suppose ̄{x} is the mean of an SRS of size $n$ drawn from a large population with mean $μ$ and standard deviation $σ$.
• Mean of the Sampling Distribution (μ_{̄{x}}):   - μ_{̄{x}} = μ   - This indicates that the sample mean ̄{x} is an unbiased estimator of the population mean $μ$.
• Standard Deviation of the Sampling Distribution (σ_{̄{x}}{}):   - σ_{̄{x}} = \frac{σ}{\sqrt{n}}   - This formula is approximately correct as long as the 10% condition is satisfied: $n < 0.10N$.   - Interpretation: The value σ_{̄{x}} measures the typical distance between a sample mean and the population mean.

Behavior of ̄{x} in Repeated Samples

• Variability Factors: The variability of ̄{x} depends on both the variability of the population ($σ$) and the sample size ($n$).   - Populations with higher variability result in more variable values of ̄{x}.   - Larger samples result in less variable values of ̄{x}.   - The Inverse Square Root Relationship: Specifically, multiplying the sample size by $4$ cuts the standard deviation of its sampling distribution in half.
• Sampling With vs. Without Replacement:   - With replacement: The standard deviation is exactly $\frac{σ}{\sqrt{n}}$.   - Without replacement: Observations are not independent, making the actual standard deviation smaller than the formula suggests. However, if the sample size is less than $10\%$ of the population size, the formula is considered nearly correct.   - Note: Larger samples provide more information. If the sample size exceeds $10\%$ of the population size, a "finite population correction" is required (though avoided in this specific text).
• Shape Consistency: These facts regarding the mean and standard deviation of ̄{x} hold true regardless of the shape of the population distribution.

AP® Exam Tip: Proper Notation

• Correct notation is vital on the AP Exam. Different symbols have specific meanings: p, ̄{x}, n, μ, μ_{̄{x}}, σ, σ_{̑{x}}, \hat{p}.
• Using incorrect notation can cause a loss of credit. If unsure, it is better to avoid the notation than to use it incorrectly.

Example Problem: Movie Attendance Statistics

• Problem Context: A large high school has a population mean of $19.3$ movies viewed in the last year with a standard deviation of $15.8$ movies.
• Sample Details: An SRS of $n = 100$ students is taken.
• Questions and Solutions:   - (a) Identify the mean of the sampling distribution:     - Solution: \u03BC_{̄{x}} = μ = 19.3\text{ movies}.   - (b) Calculate and interpret the standard deviation; verify the 10% condition:     - Condition Verification: Assume $n = 100$ is less than $10\%$ of the total students at the large high school.     - Calculation: \u03C3_{̄{x}} = \frac{15.8}{\sqrt{100}} = \frac{15.8}{10} = 1.58\text{ movies}.     - Interpretation: In SRSs of size $100$, the sample mean number of movies viewed will typically vary by about $1.58\text{ movies}$ from the true population mean of $19.3\text{ movies}$.

Mathematical Derivation of Mean and Standard Deviation (Think About It)

• The Independent Random Variable Model: Let measurements in a sample of size $n$ be $X_1, X_2, \dots, X_n$. For a large population, these are independent random variables with mean $μ$ and standard deviation $σ$.
• Definition of Sample Mean: $\u0304{x} = \frac{(X_1 + X_2 + \dots + X_n)}{n}$.
• Let $T$ be the sum: $T = X_1 + X_2 + \dots + X_n$.
• Mean Calculation for $T$ and ̄{x}:   - $\u03BC_T = μ + μ + \dots + μ = nμ$   - Since $\u0304{x} = (\frac{1}{n})T$, then \u03BC_{̄{x}} = (\frac{1}{n})μ_T = (\frac{1}{n})(nμ) = μ
• Standard Deviation Calculation for $T$ and ̄{x}:   - Using variance rules: $\u03C3_T^2 = σ^2 + σ^2 + \dots + σ^2 = nσ^2$   - Standard deviation of the sum: $\u03C3_T = \sqrt{nσ^2} = σ\sqrt{n}$   - Since $\u0304{x} = (\frac{1}{n})T$, then \u03C3_{̄{x}} = (\frac{1}{n})σ_T = \frac{σ\sqrt{n}}{n} = \frac{σ}{\sqrt{n}}

Section 7.3 Practice and Review Questions

• Exercise 54: Sample Size and Standard Deviation   - Decreasing sample size from $750$ to $375$ (halving the size) would multiply the standard deviation by:     - (a) $2$     - (b) $\sqrt{2}$     - (c) $1/2$     - (d) $1/\sqrt{2}$     - (e) none of these.
• **Exercise 55: Normality of $\hat{p}$   - The sampling distribution of $\hat{p}$ is approximately Normal because:     - (a) there are at least $7500$ Division I college athletes.     - (b) $np = 225$ and $n(1-p) = 525$ are both at least $10$.     - (c) a random sample was chosen.     - (d) the responses are quantitative.     - (e) it always has this shape.
• Exercise 56: Party Affiliation Probabilities   - In a district where $55\%$ of voters are Democrats, which expression represents the probability of getting less than $50\%$ Democrats in a sample of size $100$?   - Target probability: $P(Z < \frac{0.50 - 0.55}{\sqrt{\frac{0.55(0.45)}{100}}})$.
• Exercise 57: Music Sharing Table (Recycle and Review)   - Data provided: $29\%$ download music, $21\%$ share music, $12\%$ do both.   - (a) Create a two-way table.   - (b) What percent of users neither download nor share?   - (c) Given a user downloads music, what is the probability they also share?
• Exercise 58: Whole Grains Observational Study   - Context: People eating $3$ servings of whole grains daily have low risk of heart disease ($-20\%$), stroke, or cancer ($-15\%$).   - (a) Explain how confounding makes establishing cause-and-effect difficult.   - (b) Explain how researchers could establish a cause-and-effect relationship (suggesting an experimental design).