U.S. adults responding to a 2017 Harris poll, 47 % said they...

U.S. adults responding to a 2017 Harris poll, $47 \%$ said they always or often read nutrition labels when grocery shopping. a. Construct a $95 \%$ confidence interval for the population proportion of U.S. adults who always or often read nutrition labels when grocery shopping. b. What is the width of the $95 \%$ confidence interval? c. Name a confidence level that would produce an interval wider than the $95 \%$ confidence interval. Explain why you think this interval would be wider than a $95 \%$ confidence interval. d. Construct the interval using the confidence level you proposed in part $\mathrm{c}$ and find the width of the interval. Is this interval wider than the $95 \%$ confidence interval?

U.S. adults responding to a 2017 Harris poll, $47 \%$ said they always or often read nutrition labels when grocery shopping.
a. Construct a $95 \%$ confidence interval for the population proportion of U.S. adults who always or often read nutrition labels when grocery shopping.
b. What is the width of the $95 \%$ confidence interval?
c. Name a confidence level that would produce an interval wider than the $95 \%$ confidence interval. Explain why you think this interval would be wider than a $95 \%$ confidence interval.
d. Construct the interval using the confidence level you proposed in part $\mathrm{c}$ and find the width of the interval. Is this interval wider than the $95 \%$
confidence interval?

Key Concepts

Confidence Interval for a Population Proportion

A confidence interval for a population proportion provides a range of values within which the true proportion of the population is expected to lie, based on a sample statistic. It is constructed using the sample proportion and by accounting for the variability in the sampling process through the standard error and an appropriate critical value from the normal distribution.

Standard Error of the Proportion

The standard error of the proportion quantifies the variability of the sample proportion estimate from sample to sample. It is calculated using the formula sqrt(p?(1 - p?)/n), where p? is the sample proportion and n is the sample size. This value is critical in determining the width of a confidence interval by measuring the spread of the sampling distribution around the true proportion.

Critical Value (Z-Score)

The critical value, often denoted as Z, is the number of standard deviations from the mean required to capture the desired confidence level in a normal distribution. For a 95% confidence interval, the critical value is usually about 1.96; however, it changes with different confidence levels, affecting the margin of error and ultimately the width of the interval.

Margin of Error

The margin of error is the product of the critical value and the standard error, representing half the width of the confidence interval. It quantifies the extent of uncertainty in the estimate of the population proportion and determines how far above and below the sample proportion the true population proportion is expected to lie.

Confidence Level

The confidence level indicates the proportion of times that the confidence interval would contain the true population parameter if the same sampling procedure were repeated many times. A higher confidence level implies a greater degree of certainty that the interval contains the parameter, but it also results in a wider interval due to a larger critical value.

Width of the Confidence Interval

The width of the confidence interval is the total span between the upper and lower bounds of the interval. It is calculated as twice the margin of error. A wider interval indicates greater uncertainty in the estimation of the population parameter, often occurring when a higher confidence level is chosen.

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