When it comes to advertising, "tweens" are not ready for the...

When it comes to advertising, "tweens" are not ready for the hard-line messages that advertisers often use to reach teenagers. The Geppeto Group study $^{\star}$ found that $78 \%$ of 'tweens understand and enjoy ads that are silly in nature. Suppose that the study involved $n=1030$ 'tweens. a. Construct a $90 \%$ confidence interval for the proportion of 'tweens who understand and enjoy ads that are silly in nature. b. Do you think that "more than $75 \%$ " of all 'tweens enjoy ads that are silly in nature? Why?

When it comes to advertising, "tweens" are not ready for the hard-line messages that advertisers often use to reach teenagers. The Geppeto Group study $^{\star}$ found that $78 \%$ of 'tweens understand and enjoy ads that are silly in nature. Suppose that the study involved $n=1030$ 'tweens.
a. Construct a $90 \%$ confidence interval for the proportion of 'tweens who understand and enjoy ads that are silly in nature.
b. Do you think that "more than $75 \%$ " of all 'tweens enjoy ads that are silly in nature? Why?

Key Concepts

Hypothesis Testing for a Proportion

Hypothesis testing for a proportion is a statistical method used to assess whether observed data provide sufficient evidence to support or refute a claim about a population proportion. It involves formulating a null hypothesis, calculating a test statistic using the estimated proportion and its standard error, and comparing this statistic to a critical value or p-value threshold.

Central Limit Theorem for Proportions

The Central Limit Theorem stipulates that for a sufficiently large sample size, the distribution of the sample proportion approximates a normal distribution, regardless of the population's distribution. This justification allows for the use of z-scores when constructing confidence intervals and performing hypothesis tests for a proportion.

Standard Error of a Proportion

The standard error of a proportion measures the variability of the sample proportion around the true population proportion. It is computed using the formula that involves the sample proportion, its complement, and the sample size, thereby reflecting the precision of the estimate.

Confidence Interval for a Population Proportion

A confidence interval for a population proportion provides a range of values within which the true population proportion is likely to fall, based on a sample statistic and a specified confidence level. It takes into account the variability of the data and quantifies the uncertainty of the estimate.

Margin of Error

The margin of error is the maximum expected difference between the true population parameter and the sample estimate. In the context of a proportion, it is calculated by multiplying the standard error by the critical value from the normal distribution that corresponds to the desired confidence level.

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