In a Household Economic Survey about life satisfaction among...

In a Household Economic Survey about life satisfaction among 5849 New Zealand households, the following results were reported. Estimate with $95 \%$ confidence the proportion of all New Zealand households that are satisfied or very satisfied with their lives. $$ \begin{array}{|l|c|} \hline \text { Source } & \text { Percentage (\%) } \\ \hline \text { Very satisfied } & 26.0 \\ \hline \text { Satisfied } & 50.5 \\ \hline \begin{array}{l} \text { Neither satisfied nor } \\ \text { dissatisfied } \end{array} & 14.9 \\ \hline \text { Dissatisfied } & 6.5 \\ \hline \text { Very dissatisfied } & 2.1 \\ \hline \end{array} $$

In a Household Economic Survey about life satisfaction among 5849 New Zealand households, the following results were reported. Estimate with $95 \%$ confidence the proportion of all New Zealand households that are satisfied or very satisfied with their lives.
$$
\begin{array}{|l|c|}
\hline \text { Source } & \text { Percentage (\%) } \\
\hline \text { Very satisfied } & 26.0 \\
\hline \text { Satisfied } & 50.5 \\
\hline \begin{array}{l}
\text { Neither satisfied nor } \\
\text { dissatisfied }
\end{array} & 14.9 \\
\hline \text { Dissatisfied } & 6.5 \\
\hline \text { Very dissatisfied } & 2.1 \\
\hline
\end{array}
$$

Key Concepts

Sample Proportion

The sample proportion is the fraction of subjects in a sample exhibiting a particular characteristic. It serves as an estimate of the population proportion and is calculated by dividing the number of favorable responses by the total sample size. This concept is fundamental in survey analysis and forms the basis of further inferential procedures.

Confidence Interval for a Proportion

A confidence interval for a proportion provides a range of plausible values for the true population proportion based on the sample data. It is constructed by taking the sample proportion and adding and subtracting a margin of error, which accounts for sampling variability. This interval is interpreted to mean that there is a specified level of confidence, such as 95%, that it contains the true proportion.

Standard Error

The standard error of a proportion quantifies the variability of the sample proportion around the true population proportion. It is computed using the formula based on the sample proportion and sample size. This measure is crucial for assessing how much the sample estimate is expected to deviate from the actual population value, and it directly impacts the width of the confidence interval.

Margin of Error

The margin of error represents the radius of the confidence interval around the sample estimate. It is determined by multiplying the standard error by the critical value from a standard normal distribution (for large samples) corresponding to the desired confidence level. The margin of error provides an indication of the maximum expected difference between the true population parameter and the sample estimate.

Central Limit Theorem

The Central Limit Theorem is a key statistical principle that justifies the normal approximation of the sampling distribution of the sample proportion, especially for large sample sizes. It states that, regardless of the distribution of the underlying data, the distribution of the sample proportion will approximate a normal distribution, which enables the use of standard statistical techniques such as constructing confidence intervals.

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