Week 4: Sampling Distributions & Confidence Intervals (part 1)

Flashcards covering key concepts from Research Methods & Statistics relating to Sampling Distributions and Confidence Intervals.

Sampling error

The error associated with examining statistics calculated from a sample rather than the population.

Sample

A subset of individuals from a larger population used to estimate characteristics of the whole population.

Population

The entire group of individuals or instances about whom we hope to learn.

Sampling distribution

A distribution of a sample statistic obtained by repeatedly sampling from a population.

Central Limit Theorem

States that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Standard error

The standard deviation of the sampling distribution of the sample mean, representing how much the sample mean is expected to vary from the population mean.

Sample size

The number of observations in a sample, which affects the magnitude of sampling error.

Z-score

A statistical measurement that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations.

Mean (μ)

The average value of a population or sample.

Standard deviation (σ)

A measure of the amount of variation or dispersion in a set of values.

Confidence interval

A range of values derived from a sample statistic that is likely to contain the population parameter.

What is the formula for calculating the Standard Error (SE) of the mean?

The formula is SE = \frac{\sigma}{\sqrt{n}}, where \sigma is the population standard deviation and n is the sample size.

Under the Central Limit Theorem, what is the 'Rule of Thumb' for a sufficiently large sample size?

A sample size of n \ge 30 is generally considered large enough for the sampling distribution of the mean to be approximately normal.

Parameter vs. Statistic

A parameter is a descriptive measure of a population (e.g., \mu), whereas a statistic is a descriptive measure of a sample (e.g., \bar{x}).

Practical: If you quadruple the sample size (n), what happens to the Standard Error?

The Standard Error is halved, because it is inversely proportional to the square root of the sample size (\sqrt{4n} = 2\sqrt{n}).

Practical: If a Z-score is calculated as -1.5, what does this tell you about the data point?

It indicates the data point is 1.5 standard deviations below the mean.

How does increasing the Confidence Level (e.g., from 95\% to 99\%) impact the width of the confidence interval?

Increasing the confidence level increases the width of the confidence interval to provide a higher degree of certainty that the population parameter is captured.

Point Estimate

A single numerical value, such as the sample mean (\bar{x}), used to estimate a corresponding population parameter.

Margin of Error (ME)

The range above and below a sample statistic in a confidence interval, representing the maximum expected difference between the statistic and the parameter.

Formula for the Z-score of a Sample Mean

The formula is Z = \frac{\bar{x} - \mu}{SE} = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}.

Practical: If the population distribution is already normal, is a sample size of n \ge 30 required for the sampling distribution to be normal?

No. If the population is normally distributed, the sampling distribution of the mean will be normal regardless of the sample size.

Practical: Calculate the Standard Error (SE) where \sigma = 40 and n = 100.

SE = \frac{40}{\sqrt{100}} = \frac{40}{10} = 4.

Relation between Sample Size and Margin of Error

As the sample size (n) increases, the Margin of Error decreases, assuming the confidence level and standard deviation remain constant.

Exam Step-by-Step: Calculate the Z-score for a Sample Mean

If \bar{x} = 105, \mu = 100, \sigma = 20, and n = 25:

1. Find Standard Error: SE = \frac{20}{\sqrt{25}} = 4.

2. Calculate Z: Z = \frac{105 - 100}{4} = 1.25.

Exam Step-by-Step: Calculate a 95\% Confidence Interval

Given \bar{x} = 50, \sigma = 10, n = 100, and critical value z^* = 1.96:

1. SE = \frac{10}{\sqrt{100}} = 1.

2. ME = 1.96 \times 1 = 1.96.

3. CI = (50 - 1.96, 50 + 1.96) = (48.04, 51.96).

Exam Context: Finding Point Estimate from a given Interval

If a confidence interval is (120, 140), the point estimate (\bar{x}) is the midpoint: \frac{120 + 140}{2} = 130. The Margin of Error is half the width: 10.

Practical: Effect of Standard Deviation on Interval Width

As the population standard deviation (\sigma) increases, the Standard Error increases, which results in a wider confidence interval.

Condition Check: When can you assume the Sampling Distribution is Normal?

1. If the population is normal, the distribution of \bar{x} is normal for any n.

2. If the population is not normal, the distribution of \bar{x} is approximately normal if n \ge 30 (via CLT).

Practical: Relation between Sample Size and ME

To cut the Margin of Error in half, you must increase the sample size by a factor of four (4n).