Statistics: Descriptive & Confidence Intervals

Study Notes

Descriptive statistics summarize and describe the characteristics of a dataset, revealing patterns and trends within the data.
It provides a concise overview of the data without making inferences or generalizations about the population.
Examples of descriptive statistics include:
- Mean: The average of a dataset
- Median: The middle value in a sorted dataset
- Mode: The most frequent value in a dataset
A t-test is a statistical test used to compare the means of two groups, making it an inferential statistic rather than a descriptive statistic.

Confidence intervals provide a range of values within which we are confident the true population parameter lies.
A 95% confidence interval means that if we repeat the sampling process many times, we would expect the true population parameter to fall within that range 95% of the time.
Increasing the sample size generally leads to a narrower confidence interval.

Inferential statistics use sample data to draw conclusions about a larger population.
This branch of statistics helps us make generalizations about the entire population based on the information collected from a sample.
It plays a vital role in making informed decisions, predictions, and inferences about unknown populations.

The standard error of the mean is a measure of the variability of the sample mean.
It tells us how much the sample mean is likely to vary from the true population mean.
A smaller SEM indicates that the sample mean is a more precise estimate of the population mean.

Hypothesis testing is a statistical process used to determine whether there is enough evidence to reject a null hypothesis.
The p-value is a probability that measures the strength of evidence against the null hypothesis.
A small p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that the observed result is unlikely to have occurred by chance alone.
It's not the probability that the null hypothesis is true; instead, it's the probability of obtaining the observed result under the assumption that the null hypothesis is true.

Type I Error (False Positive): Rejecting the null hypothesis when it is actually true.
Type II Error (False Negative): Failing to reject the null hypothesis when it is actually false.
The p-value helps us make decisions about whether to reject or fail to reject the null hypothesis but does not directly represent the probability of making either a Type I or Type II error.

Descriptive statistics summarizes and describes data without making inferences about a population.
Mean, Median, and Mode are all examples of descriptive statistics.
t-test is an inferential statistical test and not a descriptive statistic.

A 95% confidence interval describes the range where we are 95% confident the true population mean lies.
The confidence interval does not guarantee that the population mean lies within the interval, but it provides a range where it is likely to exist.

Inferential statistics uses sample data to make inferences or predictions about a population.
The goal of inferential statistics is to draw conclusions about a population based on information obtained from a sample.

Increasing the sample size (N) generally leads to a narrower confidence interval.
Larger sample sizes provide more reliable information about the population, thus reducing the uncertainty and resulting in a narrower interval.

The Standard Error of the Mean (SEM) is the standard deviation of the sampling distribution of the sample mean.
Essentially, it estimates the variability of sample means around the true population mean.

The p-value in hypothesis testing represents the probability of obtaining a result as extreme as the observed result, assuming the null hypothesis is true.
A low p-value (typically below 0.05) suggests that the observed result is unlikely under the null hypothesis, leading us to reject the null hypothesis.
The p-value does not represent the probability of the null hypothesis being true, nor does it indicate the probability of making a Type I or Type II error.