Chapter 7

Chapter 7: Survey Sampling and Inference

Section 7.1: Learning About the World Through Surveys

This section introduces the foundational concepts of survey sampling, including populations, parameters, samples, and statistics. Understanding these terms is essential for interpreting survey results and making inferences about larger groups.

• Population: The entire set of individuals or objects of interest in a study.

• Parameter: A numerical summary that describes a characteristic of the population (e.g., population mean μ, population proportion p).

• Sample: A subset of the population selected for analysis.

• Statistic: A numerical summary calculated from the sample data (e.g., sample mean x̄, sample proportion p̂).

Example: In a survey of Canadian adults, the population is all Canadian adults, the parameter might be the proportion who own a cell phone, the sample is the group surveyed, and the statistic is the proportion in the sample who own a cell phone.

Notation

Concept	Population Symbol	Sample Symbol
Proportion	p	p̂
Mean	μ	x̄
Standard Deviation	σ	s

Section 7.2: Measuring the Quality of a Survey

This section discusses sources of error in surveys, focusing on bias and the importance of accuracy and precision in statistical estimation.

• Bias: Systematic error that leads to an estimate consistently off target. Types include:

  • Measurement Bias: Poorly worded questions or faulty measurement tools.

  • Sampling Bias: When the sample is not representative of the population.

• Accuracy: How close an estimate is to the true value.

• Precision: How much estimates vary if the survey is repeated.

Example: A survey that consistently overestimates the percentage of cell phone owners due to non-random sampling is biased.

Evaluating Surveys

• Consider the method of data collection and the outcome of a survey.

• Random sampling reduces bias and increases the reliability of inferences.

Section 7.3: The Central Limit Theorem for Sample Proportions

The Central Limit Theorem (CLT) is a fundamental result that describes the behavior of sample proportions. It allows us to use normal probability methods for inference when certain conditions are met.

• Sampling Distribution of p̂: The probability distribution of the sample proportion over all possible samples of a fixed size from the population.

• Mean of p̂: $\mu_{\hat{p}} = p$

• Standard Deviation of p̂: $\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$

Conditions for the CLT (Sample Proportions)

• Random sample

• Sample size is less than 10% of the population (independence)

• Large enough sample: $np \geq 10$ and $n(1-p) \geq 10$

CLT Conclusions

• If conditions are met, the sampling distribution of p̂ is approximately normal:

$\hat{p} \sim N\left(p, \sqrt{\dfrac{p(1-p)}{n}}\right)$

Example: Estimating Probabilities with the CLT

Suppose 34% of Canadian households own a dog. In a random sample of 200 households, the probability that more than 40% own a dog can be estimated using the normal approximation for p̂.

Section 7.4: Estimating the Population Proportion with Confidence Intervals

Confidence intervals provide a range of plausible values for a population parameter, such as a proportion, based on sample data.

• Confidence Interval for a Proportion:

$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

• Margin of Error (E): $E = z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

• Confidence Level: The probability that the interval contains the true parameter in repeated samples (commonly 90%, 95%, or 99%).

Table: Common z* Values for Confidence Levels

Confidence Level	z*
90%	1.645
95%	1.96
99%	2.576

Example: Constructing a Confidence Interval

In a sample of 800 adults, 352 own a tablet. The sample proportion is $\hat{p} = 352/800 = 0.44$. A 95% confidence interval is:

$0.44 \pm 1.96 \sqrt{\dfrac{0.44 \times 0.56}{800}}$

Interpreting the Confidence Interval

• If the survey were repeated many times, about 95% of the calculated intervals would contain the true population proportion.

Section 7.5: Margin of Error and Sample Size for Proportions

This section explains how to determine the sample size needed to achieve a desired margin of error for a given confidence level.

• Sample Size Formula for Desired Margin of Error (E):

$n = \dfrac{(z^*)^2 \hat{p}(1-\hat{p})}{E^2}$

• If no prior estimate for $\hat{p}$ is available, use $\hat{p} = 0.5$ for the most conservative (largest) sample size.

Example: Planning a Poll

To estimate a proportion with 95% confidence and a margin of error of 0.03, the required sample size is:

$n = \dfrac{(1.96)^2 \times 0.5 \times 0.5}{0.03^2} = 1068$

Effect of Confidence Level and Margin of Error

• Higher confidence level → wider interval, larger sample size needed.

• Smaller margin of error → larger sample size needed.

Summary Table: Effects of Confidence Level and Margin of Error

Change	Effect on Interval Width	Effect on Sample Size
Increase Confidence Level	Wider	Larger
Decrease Margin of Error	Narrower	Larger

Key Takeaways

• Random sampling and proper survey design are essential for valid inference.

• The Central Limit Theorem allows normal approximation for sample proportions under certain conditions.

• Confidence intervals quantify uncertainty in estimates of population proportions.

• Sample size calculations ensure desired precision in survey results.

Additional info: These notes expand on the slides by providing definitions, formulas, and examples for each concept, ensuring a self-contained study guide for exam preparation.