Sampling Distributions

It doubles MOE since it increases variability by adding more data to one side only.

It halves MOE because it provides more information about one population's true proportion value.

It decreases MOE less than if both were doubled but more than if neither were changed.

Doubling one sample size has no effect on MOE as changes need to be symmetrical across both samples to impact results significantly.

Sample mean (x̄)

Percentile rank

Difference in sample proportions ($\hat{p}_1 - \hat{p}_2$)

Standard error (SE)

Ensuring similarity of practice exam difficulty across all participants.

Random assignment of students to either studying technique.

Random selection from among teachers willing to incorporate techniques.

Maintaining consistency in timing and conditions under which exams are taken.

There may have been an error in constructing the confidence interval or calculating the point estimate.

The population proportion must have changed since taking the samples.

This is expected due to random variability and does not imply any error or significance.

Their samples are likely too small to produce a reliable estimate of population differences.

The distribution of the sample means from each city.

The distribution of possible values for the difference between the sample proportions if the study were repeated many times.

The distribution of the difference between the population proportions in the two cities.

The distribution of the sample proportions from each city.

The estimated proportion of voters might be very similar across both regions studied.

One study must have sampled incorrectly because perfect similarity is highly unlikely.

The smaller variance observed necessitates larger underlying populations.

Different demographic factors were likely controlled differently across studies resulting in misleading similarities.

The tests conducted are likely invalid due to unaccounted-for changes within populations or methods year over year.

There might, has been an increase in likelihood of change over time resulting from cumulatively repeated testing.

Given the same testing conditions, there should be consistency in p-values obtained each year, regardless of actual population changes.

The evidence against null hypothesis suggesting no change has strengthened compared to last year's findings

Each distribution would look identical regardless of differing true population proportions since large samples homogenize sampling distributions

The spreads among distributions vary greatly despite large samples because true population proportions' influence outweighs benefits provided by large samples

All distributions would converge into one single distribution as they reflect combined characteristics from differing underlying populations proportionately

Each distribution would center around its own population proportion but have similar shapes and spreads due to large samples providing consistency across distributions’ variability

The t-distribution method, taking into account degrees freedom estimated separately each.

Apply Normal Distribution directly after ensuring each sample's mean is above minimum size requirement.

Use z-Distribution with pooled standard deviation estimates.

Perform Bootstrapping Techniques in order to generate replication data sets for interval calculation.

Sample proportion

Population proportion

Margin of error for proportion estimate

Test statistic value