In a simple random sample of 1200 Americans age 20 and over,...

In a simple random sample of 1200 Americans age 20 and over, the proportion with diabetes was found to be $0.115$ (or $11.5 \%)$. a. What is the standard error for the estimate of the proportion of all Americans age 20 and over with diabetes? b. Find the margin of error, using a $95 \%$ confidence level, for estimating this proportion. c. Report the $95 \%$ confidence interval for the proportion of all Americans age 20 and over with diabetes. d. According to the Centers for Disease Control and Prevention, nationally, $10.7 \%$ of all Americans age 20 or over have diabetes. Does the confidence interval you found in part c support or refute this claim? Explain.

In a simple random sample of 1200 Americans age 20 and over, the proportion with diabetes was found to be $0.115$ (or $11.5 \%)$.
a. What is the standard error for the estimate of the proportion of all
Americans age 20 and over with diabetes?
b. Find the margin of error, using a $95 \%$ confidence level, for estimating this proportion.
c. Report the $95 \%$ confidence interval for the proportion of all Americans
age 20 and over with diabetes.
d. According to the Centers for Disease Control and Prevention, nationally, $10.7 \%$ of all Americans age 20 or over have diabetes. Does the confidence interval you found in part c support or refute this claim? Explain.

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Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from the sample data and the margin of error, that is believed to cover the true population parameter with a specified level of confidence. It provides a quantitative measure of the precision of a sample estimate and helps in making inferences about the population.

Standard Error

The standard error measures how much the sample proportion is expected to vary from one random sample to another due to random chance. It provides an estimate of the variability inherent in the sampling process and is crucial for constructing confidence intervals and conducting hypothesis tests.

Margin of Error

The margin of error represents the extent of the uncertainty around the sample estimate at a given confidence level. It is calculated by multiplying the standard error by the critical value associated with the desired level of confidence, and it indicates the range within which the true population parameter is likely to fall.

Interpretation of Confidence Interval

Interpreting a confidence interval involves analyzing whether a specific value or claim about the population parameter lies within the interval. This interpretation is essential for validating hypotheses or claims about the population based on the sample data.

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