An insurance company checks police records on 567 accidents...

An insurance company checks police records on 567 accidents selected at random and notes that teenagers were at the wheel in 86 of them. a) Create a $95 \%$ confidence interval for the percentage of all auto accidents that involve teenage drivers. b) Explain what your interval means. c) Explain what "95\% confidence" means. d) A politician urging tighter restrictions on drivers' licenses issued to teens says, "In one of every five auto accidents, a teenager is behind the wheel." Does your confidence interval support or contradict this statement? Explain.

An insurance company checks police records on 567 accidents selected at random and notes that teenagers were at the wheel in 86 of them.
a) Create a $95 \%$ confidence interval for the percentage of all auto accidents that involve teenage drivers.
b) Explain what your interval means.
c) Explain what "95\% confidence" means.
d) A politician urging tighter restrictions on drivers' licenses issued to teens says, "In one of every five auto accidents, a teenager is behind the wheel." Does your confidence interval support or contradict this statement? Explain.

Key Concepts

Confidence Interval for a Population Proportion

A confidence interval for a population proportion provides a range of plausible values for an unknown proportion based on sample data. It uses the sample proportion as a point estimate and incorporates the variability in the data through the standard error, yielding an interval that, under repeated sampling, is expected to contain the true proportion a specified percentage of the time.

Margin of Error

The margin of error is the product of a critical value (often derived from the normal distribution for large samples) and the standard error of the estimate. It quantifies the maximum expected difference between the sample estimate and the true population parameter, and is key to determining the width of a confidence interval.

Interpretation of Confidence Levels

The confidence level, such as 95%, represents the long-run proportion of confidence intervals that would contain the true population parameter if the same sampling procedure were repeated many times. It does not imply that there is a 95% probability that the particular computed interval contains the parameter, but rather that the method used to calculate the interval is reliable in the long run.

Sampling Distribution of the Proportion

The sampling distribution of the proportion describes the distribution of sample proportions that would arise from many random samples of the same size from the population. For large sample sizes, this distribution can be approximated by a normal distribution, which justifies the use of the standard error and z-scores in constructing confidence intervals.

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