Speeding Drivers. In 2017, ABC News reported that 58 % of...

Speeding Drivers. In 2017, ABC News reported that $58 \%$ of U.S. drivers admit to speeding. Suppose that a new satellite technology can instantly measure the speed of any vehicle on a U.S. road and determine whether the vehicle is speeding, and this satellite technology was used to take a sample of 20,000 vehicles at $6: 00$ p.m. EST on a recent Tuesday afternoon. Of these 20,000 vehicles, 9,252 were speeding. a. What is the sample proportion of vehicles on U.S. roads that speed? b. Using $99 \%$ confidence, what is the margin of error? c. Using the results from parts (a) and (b), develop the $99 \%$ confidence interval estimate of the proportion of vehicles on U.S. roads that speed. d. What does the confidence interval in part (c) lead you to conclude about the ABC News report?

Speeding Drivers. In 2017, ABC News reported that $58 \%$ of U.S. drivers admit to speeding. Suppose that a new satellite technology can instantly measure the speed of any vehicle on a U.S. road and determine whether the vehicle is speeding, and this satellite technology was used to take a sample of 20,000 vehicles at $6: 00$ p.m. EST on a recent Tuesday afternoon. Of these 20,000 vehicles, 9,252 were speeding.
a. What is the sample proportion of vehicles on U.S. roads that speed?
b. Using $99 \%$ confidence, what is the margin of error?
c. Using the results from parts (a) and (b), develop the $99 \%$ confidence interval estimate of the proportion of vehicles on U.S. roads that speed.
d. What does the confidence interval in part (c) lead you to conclude about the ABC News report?

Key Concepts

Sample Proportion

The sample proportion is the fraction of the sample that exhibits a specific characteristic, calculated by dividing the number of favorable outcomes by the total sample size. It serves as an estimator for the true population proportion and is fundamental in inferential statistics, forming the basis for further analysis such as establishing confidence intervals.

Margin of Error

The margin of error quantifies the extent of the potential difference between the sample estimate and the true population parameter due to random sampling variability. It is computed using the standard error of the estimate and the appropriate critical value from the standard normal (or t-) distribution for the desired confidence level, thereby defining the range within which the true proportion is estimated to lie.

Confidence Interval

A confidence interval is a range of values, derived from the sample statistic and margin of error, used to estimate the true population parameter with a specified level of confidence. It represents the interval within which the true proportion is likely to fall, with the confidence level indicating the probability that the interval constructed from repeated samples would capture the true parameter.

Interpretation of Confidence Interval

Interpreting a confidence interval involves assessing whether a certain hypothesized or reported value falls within the estimated range of the population parameter. When evaluating claims, such as a reported national proportion, if the reported value lies outside the confidence interval, it suggests that the actual proportion might differ from the claim, thereby prompting further investigation or questioning the validity of the original report.

Transcript

Please give Ace some feedback