Social Media Usage. The Pew Research Center has conducted...

Social Media Usage. The Pew Research Center has conducted extensive research on social media usage. One finding, reported in June 2018 , was that $78 \%$ of adults aged 18 to 24 use Snapchat. Another finding was that $45 \%$ of those aged 18 to 24 use Twitter. Assume the sample size associated with both findings is 500. a. Develop a $95 \%$ confidence interval for the proportion of adults aged 18 to 24 who use Snapchat. b. Develop a $99 \%$ confidence interval for the proportion of adults aged 18 to 24 who use Twitter. c. In which case, part (a) or part (b), is the margin of error larger? Explain why.

Social Media Usage. The Pew Research Center has conducted extensive research on social media usage. One finding, reported in June 2018 , was that $78 \%$ of adults aged 18 to 24 use Snapchat. Another finding was that $45 \%$ of those aged 18 to 24 use Twitter. Assume the sample size associated with both findings is 500.
a. Develop a $95 \%$ confidence interval for the proportion of adults aged 18 to 24 who use Snapchat.
b. Develop a $99 \%$ confidence interval for the proportion of adults aged 18 to 24 who use Twitter.
c. In which case, part (a) or part (b), is the margin of error larger? Explain why.

Key Concepts

Confidence Interval for Proportions

A confidence interval for proportions is a range of values derived from sample data that is likely to contain the true population proportion with a specified level of confidence (such as 95% or 99%). It is calculated using the sample proportion, the standard error, and a critical value from the standard normal distribution, encapsulating the estimator's variability and uncertainty based on the sample size.

Standard Error

The standard error measures the variability or deviation of the sample proportion from the true population proportion. It is computed using the formula ?[p(1-p)/n], where p is the sample proportion and n is the sample size. This value indicates how much the proportion is expected to fluctuate from sample to sample, and is crucial in constructing the confidence interval.

Margin of Error

The margin of error represents the maximum expected difference between the true population proportion and the sample estimate, given a specified confidence level. It is calculated by multiplying the standard error by the critical value (z-score) corresponding to the desired level of confidence. A larger margin of error indicates a wider interval and less precision in the estimate.

Critical Value (z-score)

The critical value, often represented by z, is the number of standard deviations a data point is from the mean, corresponding to a given confidence level. For example, a 95% confidence level typically uses a z-score of approximately 1.96, while a 99% confidence level uses about 2.576. The z-score increases with higher confidence levels, resulting in a wider confidence interval.

Relationship Between Confidence Level and Margin of Error

Generally, as the confidence level increases, the critical value used in the calculation of the margin of error also increases. This leads to a larger margin of error, meaning that more confidence in capturing the true population parameter results in a wider confidence interval. This concept is applicable in comparing the two parts of the question, where the 99% interval (with a higher z-score) will have a larger margin of error than the 95% interval.

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