A Harris poll asked Americans in 2016 and 2017 if they were...

A Harris poll asked Americans in 2016 and 2017 if they were happy. In $2016,31 \%$ reported being happy and in 2017 , $33 \%$ reported being happy. Assume the sample size for each poll was 1000 . A $95 \%$ confidence interval for the difference in these proportions $p_{1}-p_{2}$ (where proportion 1 is proportion happy in 2016 and proportion 2 is the proportion happy in 2017) is $(-0.06,0.02)$. Interpret this confidence interval. Does the interval contain $0 ?$ What does this tell us about happiness among American in 2016 and $2017 ?$

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Key Concepts

Confidence Interval

A confidence interval provides a range of plausible values for a population parameter, constructed such that if the same procedure is repeated multiple times, a specified percentage of the intervals will contain the true parameter. In this context, the 95% confidence interval for the difference in proportions gives a range of values that are consistent with the observed data from the polls.

Difference in Proportions

The difference in proportions refers to the subtraction of one proportion from another—in this case, the proportion of happy individuals in 2016 minus the proportion in 2017. This metric is used to assess whether there is a meaningful change or difference between the two groups being compared.

Statistical Significance and Containing Zero

When a confidence interval for a difference in proportions contains zero, it indicates that there is no statistically significant difference between the two groups at the chosen confidence level. Here, since the interval (-0.06, 0.02) includes zero, it suggests that the observed difference in happiness rates between 2016 and 2017 might be due to natural sampling variability rather than a true change in happiness levels.

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