Glossary
Alternative Hypothesis (Hₐ)
The statement that the researcher is trying to find evidence for, suggesting an effect, difference, or change from the null hypothesis.
Example:
If the null hypothesis states a drug has no effect, the alternative hypothesis (Hₐ) might be that the drug does reduce recovery time.
Confidence Intervals
A range of values, calculated from sample data, that is likely to contain the true value of a population parameter with a certain level of confidence.
Example:
A 95% confidence interval for the proportion of students who prefer online learning might be (0.60, 0.68), suggesting the true proportion is likely within this range.
Confidence Level (C)
The probability that the method used to construct the confidence interval will produce an interval that contains the true population parameter.
Example:
A 95% confidence level means that if we repeated the sampling process many times, about 95% of the resulting intervals would capture the true population proportion.
Critical Value (z*)
The number of standard deviations a sample statistic is from the mean of the sampling distribution, used to determine the margin of error for a confidence interval.
Example:
For a 95% confidence interval for proportions, the critical value (z*) is typically 1.96, derived from the standard normal distribution.
Independence (Condition for Inference)
The condition that individual observations in a sample are independent of each other, and if sampling without replacement, the sample size is less than 10% of the population size.
Example:
When surveying students about their favorite subject, it's important that one student's answer doesn't influence another's, ensuring independence of responses.
Inference with Two Proportions
Statistical methods used to compare two population proportions based on data from two independent samples.
Example:
A study comparing the success rate of a new teaching method versus a traditional method would use inference with two proportions to see if there's a significant difference.
Normality (Condition for Inference)
The condition that the sampling distribution of the statistic is approximately normal, typically checked for proportions by ensuring np ≥ 10 and n(1-p) ≥ 10.
Example:
To use a Z-interval for proportions, we check the normality condition by verifying that both the number of successes and failures in the sample are at least 10.
Null Hypothesis (H₀)
The statement of no effect, no difference, or no change, which is assumed to be true until there is sufficient evidence to reject it.
Example:
In a study testing a new drug, the null hypothesis (H₀) would be that the drug has no effect on recovery time.
P-value
The probability of observing a sample statistic as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true.
Example:
A p-value of 0.03 means there's a 3% chance of seeing our results (or more extreme) if the null hypothesis were actually true, suggesting evidence against the null.
Random Sample
A sample in which every individual in the population has an equal chance of being selected, ensuring the sample is representative and reduces bias.
Example:
To survey student opinions, a school uses a computer program to select 200 student IDs completely at random, ensuring each student has an equal chance of being chosen.
Randomness (Condition for Inference)
The condition that data must come from a well-designed random sample or randomized experiment to ensure valid statistical inference.
Example:
Before constructing a confidence interval for average height, one must confirm that the participants were selected using randomness, such as a simple random sample.
Sample Proportion (p̂)
The proportion of successes observed in a sample, calculated as the number of successes divided by the sample size. It's the best point estimate for the true population proportion.
Example:
If 55 out of 100 surveyed students prefer chocolate ice cream, the sample proportion (p̂) is 0.55.
Sample Size (n)
The total number of individuals or observations included in a sample. A larger sample size generally leads to more precise estimates.
Example:
In a study of voter preferences, a sample size of 1500 voters was surveyed to get a more accurate estimate of the population's opinion.
Significance Tests
A formal procedure used to evaluate the strength of evidence against a null hypothesis concerning a population parameter.
Example:
A researcher performs a significance test to determine if a new fertilizer significantly increases crop yield compared to the old one.
Standard Error
An estimate of the standard deviation of a sampling distribution, indicating the typical distance a sample statistic is from the true population parameter.
Example:
When calculating a confidence interval, the standard error quantifies the variability of the sample proportion, helping determine the margin of error.
Statistical Inference
The process of using data from a sample to make conclusions or predictions about a larger population.
Example:
A political pollster uses data from a random sample of 1000 voters to infer the likely outcome of an election for the entire country.