Distribution of Sample Means and Sampling Fundamentals

Flashcards reviewing key concepts from Lecture 8 and Lecture 9, including probability calculations, sampling techniques, parameters vs statistics, the 1936 presidential election polls, distribution of sample means, and standard error.

What is the distribution of sample means?

The distribution of sample means (or sampling distribution) is the distribution of means obtained when taking an infinite number of samples of a specific size $n$ from a population and plotting each sample's mean on a histogram.

Why is Nello's Place average rating (127 reviews) considered more trustworthy than The Braided Maine average rating (5 reviews), even though both have 4 stars?

Larger sample sizes reduce the impact of individual outlier reviews by chance, offering a more consistent and trustworthy estimate of the true population parameter.

What are the three distinct distributions that must be differentiated in statistics?

1. The population distribution (values for everyone in the population).
2. The sample distribution (values for a specific sample of size $n$).
3. The distribution of sample means (the collection of means from all possible samples of size $n$).

How does the mean of the distribution of sample means relate to the population mean?

The mean of the distribution of sample means is always equal to the population mean ($\mu$).

How does increasing the sample size $n$ affect the spread of the distribution of sample means?

As sample size $n$ increases, the sample means cluster more tightly around the population mean, resulting in less spread and less error between sample statistics and the population parameter.

What shape does the distribution of sample means take as sample size increases?

The distribution of sample means approaches a normal distribution, even if the underlying raw scores or population distribution are skewed or not normally distributed.

What is the standard error ($SE$) of the mean, and how is it calculated?

Standard error ($SE$) is the standard deviation of the distribution of sample means, measuring how much sample means tend to vary from the population mean. It is calculated as $SE = \frac{SD}{\sqrt{n}}$, where $SD$ is the population standard deviation and $n$ is the sample size.

What percentage of sample means fall within one standard error of the mean of the distribution of sample means?

Approximately $68\%$ of all sample means of a given size fall within one standard error above and one standard error below ($\pm 1\,SE$) the mean of the distribution of sample means.

In the London 2012 Olympics athlete height example ($\mu = 69.65$\,inches, $SD = 4.45$\,inches), what is the standard error for a sample size of $n = 4$?

The standard error is $SE = \frac{4.45}{\sqrt{4}} = \frac{4.45}{2} = 2.23$\,inches. Sample means are expected to vary around the population mean by about $\pm 2.23$\,inches.

In the London 2012 Olympics athlete height example ($\mu = 69.65$\,inches, $SD = 4.45$\,inches), what is the standard error for a sample size of $n = 100$?

The standard error is $SE = \frac{4.45}{\sqrt{100}} = \frac{4.45}{10} = 0.45$\,inches.

How is probability defined in terms of long-term repetitions?

Probability is the proportion of times a specific event occurs in a very long series of repetitions.

What is the formula for calculating probability when all outcomes are equally likely?

$$P = \frac{\text{number of ways to get the desired event}}{\text{total number of possible events}}$$

What are the minimum and maximum possible numerical values for a probability?

Probabilities range strictly between $0$ (impossible event) and $1$ (event that is certain to occur).

What is the probability of drawing a jack from a standard deck of $52$ playing cards?

Since there are $4$ jacks in a deck of $52$ cards, the probability is $P = \frac{4}{52} \approx 7.7\%$.

What is the distinction between a population and a sample?

A population is the entire group of cases a researcher wants to learn about, whereas a sample is a subset of the population from which data is actually collected.

What is the difference between a parameter and a statistic?

A parameter is a numerical summary describing a population (typically unknown), while a statistic is a numerical summary calculated from sample data used to estimate the population parameter.

How does a simple random sample differ from a convenience sample?

In a simple random sample, every case in the population has an equal probability of selection, yielding a representative sample. A convenience sample selects cases based on ease of access, introducing potential sampling bias.

Why did the 1936 Literary Digest poll fail to correctly predict the presidential election despite its large sample size?

The Literary Digest poll suffered from sampling bias by selecting participants from phone books and club memberships, which overrepresented wealthier individuals who favored Roosevelt's opponent and omitted poorer voters.

What prediction did the 1936 Gallup poll make regarding Franklin D. Roosevelt, and how did it compare to the actual parameter?

The Gallup poll surveyed $50{,}000$ participants using a representative sample and predicted Roosevelt would win with $56\%$ of the vote, close to the actual population parameter of $62\%$.