Glossary
2-PropZInt (Calculator Shortcut)
A function on graphing calculators (like TI-83/84) that automates the calculation of a two-sample z-interval for proportions.
Example:
Instead of manually calculating the standard error and critical value, I can use the 2-PropZInt function on my calculator to quickly find the confidence interval.
Categorical Variable
A variable that places individuals into one of several groups or categories. Proportions are calculated for these types of variables.
Example:
Whether a student passes or fails an exam is a categorical variable, as it places them into one of two distinct categories.
Confidence Interval
A range of plausible values for an unknown population parameter, constructed from sample data with a specified level of confidence.
Example:
After surveying students, we constructed a confidence interval to estimate the true average amount of time students spend on homework per night.
Confidence Level
The probability that the confidence interval constructed will contain the true population parameter if the process were repeated many times.
Example:
A 90% confidence level means that if we were to construct many such intervals, about 90% of them would capture the true difference in proportions.
Critical Value (z*)
The z-score that defines the boundary of the confidence interval, determined by the chosen confidence level. It indicates how many standard errors away from the point estimate the interval extends.
Example:
For a 95% confidence interval, the critical value (z*) is typically 1.96, meaning the interval extends 1.96 standard errors in each direction.
Expected Successes/Failures
The number of successes (n*p̂) and failures (n*(1-p̂)) expected in a sample, used to check the Normality (Large Counts) condition for proportions.
Example:
For a sample of 100 students where 70 passed, the expected successes are 70 and expected failures are 30, both of which must be at least 10.
Independence (10% Condition, Random Assignment)
A condition ensuring that observations within and between samples are independent. This is checked by the 10% condition (population is at least 10 times the sample size) or by random assignment in experiments.
Example:
When sampling students without replacement, we check the Independence condition by ensuring the school population is at least 10 times our sample size of 50 students.
Independent Populations
Two populations are independent if the selection of individuals from one population does not affect the selection of individuals from the other population.
Example:
Comparing the proportion of voters who support a candidate in California versus the proportion in Texas involves two independent populations.
Interpretation (of Confidence Interval)
Explaining what a confidence interval means in the context of the problem, stating the confidence level and the estimated range for the true parameter.
Example:
A correct interpretation of a confidence interval for the difference in proportions might be: 'We are 95% confident that the true difference in passing rates between Method A and Method B is between -0.094 and 0.094.'
Margin of Error
The range of values above and below the point estimate in a confidence interval, indicating the precision of the estimate.
Example:
If a survey reports a 5% margin of error, it means the true population proportion is likely within 5 percentage points of the sample proportion.
Normality (Large Counts Condition)
A condition that ensures the sampling distribution of the difference in sample proportions is approximately normal. This is met if both samples have at least 10 expected successes and 10 expected failures.
Example:
Before constructing a confidence interval for proportions, we verify Normality by checking that np̂ and n(1-p̂) are both at least 10 for each sample.
Point Estimate
A single value calculated from sample data that is used to estimate an unknown population parameter.
Example:
The difference between the two sample proportions (p̂1 - p̂2) serves as the point estimate for the true difference in population proportions.
Population Proportions
The true proportion of individuals in an entire population that possess a certain characteristic. These are typically unknown and estimated using sample data.
Example:
We might want to compare the population proportion of all teenagers who own a smartphone versus the population proportion of all adults who own one.
Randomness (Random Sampling)
A condition requiring that samples are selected randomly from their respective populations to ensure they are representative and allow for generalization.
Example:
To compare opinions on a new school policy, surveying every 10th student entering the building ensures randomness in the sample selection.
Sample Proportions (p̂1, p̂2)
The proportion of successes observed in a specific sample from a population. It serves as the point estimate for the unknown population proportion.
Example:
If 60 out of 100 surveyed students prefer pizza, the sample proportion (p̂) for pizza preference is 0.60.
Standard Error (SE)
A measure of the variability or typical distance between a sample statistic (like the difference in sample proportions) and the true population parameter.
Example:
A small Standard Error suggests that our sample difference is likely very close to the true difference between the two population proportions.
Two-Sample Z-Interval
A confidence interval used to estimate the true difference between two population proportions of a categorical variable from two independent groups.
Example:
To determine if the proportion of high school students who prefer online learning is significantly different from the proportion of college students, you would construct a Two-Sample Z-Interval.