Study sheet: Sampling Distributions and Confidence Intervals

📋 Course Outline

1. Parameters and Sample Statistics
2. Sampling Distribution of the Mean
3. Central Limit Theorem
4. Mean Sampling Distribution Applications
5. Sampling Distribution of a Proportion
6. Statistical Estimation Methods
7. Confidence Levels and Intervals
8. Confidence Interval for a Mean
9. Confidence Interval for a Proportion
10. Margin of Error and Sample Size

📖 1. Parameters and Sample Statistics

🔑 Key Concepts & Definitions

• Population parameter : a numerical descriptive measure of a population, usually unknown because it is based on all population observations.
• Sample statistic : a numerical descriptive measure calculated from the observations in a sample.

📝 Essential Points

📌 The population mean is denoted by μ and estimated by the sample mean x̄, while the population proportion is denoted by p and estimated by the sample proportion p̂.

💡 Memory Hook

A parameter describes the population, whereas a statistic describes the sample.

📖 2. Sampling Distribution of the Mean

🔑 Key Concepts & Definitions

• Sampling distribution : the probability distribution of that statistic across all possible samples of a fixed size n.

★ Must-know

📐 Formula — For a population variable with mean μ and standard deviation σ, the sampling distribution of the mean has mean $E(\bar{x})=\mu$ and standard deviation $SD(\bar{x})=\frac{\sigma}{\sqrt{n}}$, called the standard error.

Further detail

• In the ESCP weight example, the population mean is 63 kg, the population standard deviation is 23 kg, and samples contain 100 students, giving a standard error of 2.3 kg.

💡 Memory Hook

Larger samples cause smaller standard errors.

📖 3. Central Limit Theorem

★ Must-know

📌 If the original variable is normally distributed, the sampling distribution of the mean is normally distributed for any sample size.

📌 The Central Limit Theorem states that the mean of a random sample has a sampling distribution that can be approximated by a normal model, with a better approximation as n increases.

Further detail

• The sample size is usually considered large enough for the Central Limit Theorem when n≥30, although the required size depends on the shape of the sampled distribution.

💡 Memory Hook

Non-normal population → larger n → approximately normal sample means

📖 4. Mean Sampling Distribution Applications

★ Must-know

• For a population with mean 172 cm and standard deviation 10 cm, the sampling distribution of means for samples of size 50 has standard deviation $10/\sqrt{50}\approx1.41$ cm.

Further detail

• For samples of 50 heights from that population, the probability of a sample mean of at least 184 cm is approximately 0 because its z-score is 8.48.

• For the BMI example, a sample mean of 140 pounds has z-score −1.03 and probability 15.15%, so it is low but not exceptionally unusual.

📖 5. Sampling Distribution of a Proportion

🔑 Key Concepts & Definitions

• Sampling distribution of a proportion : the probability distribution of p̂ across all possible samples of size n.

★ Must-know

📌 The sampling distribution of p̂ is approximately normal when $np\ge15$ and $n(1-p)\ge15$.

Further detail

• For a budget supported by 52% of voters and a sample of 400, the standard error of p̂ is 0.025 because $\sqrt{(0.52)(0.48)/400}=0.025$.

💡 Memory Hook

Population proportion p versus sample proportion p̂.

📖 6. Statistical Estimation Methods

🔑 Key Concepts & Definitions

• Target parameter : the unknown population parameter that an estimation procedure aims to estimate, such as μ or p.
• Point estimator : a rule or formula that uses sample data to calculate a single estimate of a population parameter.
• Confidence interval : a formula that uses sample data to calculate a range of values estimating a target population parameter.

💡 Memory Hook

Point estimation gives one value; interval estimation gives a range.

📖 7. Confidence Levels and Intervals

🔑 Key Concepts & Definitions

• Confidence level : the long-run proportion of confidence intervals constructed by the method that contain the true population parameter.

★ Must-know

📐 Formula — The general confidence-interval formula is $\text{Point Estimate}\pm(\text{Critical Value})(\text{Standard Error})$.

📌 Increasing the confidence level requires a wider interval, creating a trade-off in which greater certainty produces lower precision.

Further detail

📌 A 95% confidence procedure produces intervals that contain the true parameter in 95% of repeated samples and miss it in 5% of repeated samples.

💡 Memory Hook

Higher confidence → wider interval → lower precision.

📖 8. Confidence Interval for a Mean

★ Must-know

📐 Formula — When σ is known, a confidence interval for μ is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$.

📐 Formula — When σ is unknown, a confidence interval for μ is $\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$, with degrees of freedom $df=n-1$.

📌 The z-interval for μ requires a random sample, known population standard deviation σ, and a large sample size n≥30 to support an approximately normal sampling distribution.

Further detail

• The Student t-distribution is bell-shaped and symmetric with fatter tails than the standard normal distribution, and it approaches the normal distribution as n increases.

💡 Memory Hook

Known σ uses z; unknown σ uses t.

📖 9. Confidence Interval for a Proportion

★ Must-know

📐 Formula — A confidence interval for a population proportion is $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

📌 A confidence interval for p requires a random or representative sample and sufficiently large expected counts, with $n\hat{p}\ge15$ and $n(1-\hat{p})\ge15$.

Further detail

• In the 2010 IPSOS survey, 53% of 1011 voters were satisfied with Bertrand Delanoë, producing a 95% confidence interval from 49.9% to 56.1%.

📖 10. Margin of Error and Sample Size

🔑 Key Concepts & Definitions

• Margin of error : the amount added to and subtracted from the point estimate to form a confidence interval, representing estimation imprecision.

★ Must-know

📐 Formula — The confidence interval can therefore be written as $\text{Point Estimate}\pm\text{ME}$, where $\text{ME}=\text{Critical Value}\times\text{Standard Error}$.

📌 For a fixed confidence level and variability, achieving a smaller margin of error requires a larger sample size.

Further detail

• The most commonly used confidence levels are 90%, 95%, and 99%, with normal critical values 1.645, 1.96, and 2.576 respectively.

💡 Memory Hook

Smaller margin of error requires a larger sample.

📊 Synthesis Tables

Mean Confidence Intervals

Situation	Distribution	Interval
σ known	Standard normal z	x̄ ± zα/2 σ/√n
σ unknown	Student t, df=n−1	x̄ ± tα/2 s/√n