A market researcher wants to know the percent of households...

A market researcher wants to know the percent of households in a rural area that have no vehicle. In a random sample of 120 households, 6 households had no vehicle. (a) Construct a $90 \%$ confidence interval for the percent of households in the rural area that have no vehicle. (b) Give the margin of error $E$ for the $90 \%$ confidence interval.

A market researcher wants to know the percent of households in a rural area that have no vehicle. In a random sample of 120 households, 6 households had no vehicle.
(a) Construct a $90 \%$ confidence interval for the percent of households in the rural area that have no vehicle.
(b) Give the margin of error $E$ for the $90 \%$ confidence interval.

Key Concepts

Confidence Interval for Population Proportion

A confidence interval for a population proportion is a range of values derived from sample data that is likely to contain the true proportion of a characteristic in the population. This interval is built using the sample proportion along with a margin of error that accounts for variability due to random sampling. It provides an estimate that, with a specified level of confidence, the true parameter falls within this range.

Margin of Error

The margin of error represents half the width of the confidence interval and quantifies the level of uncertainty in the estimate of the population proportion. It is determined by multiplying a critical value (commonly a Z-score corresponding to the chosen confidence level) by the standard error of the proportion. A smaller margin of error indicates a more precise estimate.

Standard Error of Proportion

The standard error of the proportion is a measure of the variability of the sample proportion from sample to sample and is given by the formula ?[p?(1 - p?)/n], where p? is the observed sample proportion and n is the sample size. This concept is essential in quantifying the precision of the sample estimate and is a key component in constructing confidence intervals.

Z-Score and Confidence Level

The Z-score is a value from the standard normal distribution that corresponds to a specified confidence level in interval estimation. When constructing a confidence interval, the Z-score serves as a multiplier in the calculation of the margin of error, ensuring that the interval is wide enough to capture the true population proportion with the desired level of certainty.