Question

In a survey of 2255 randomly selected US adults (age 18 or older), 1787 of them use the Internet regularly. Of the Internet users, 1054 use a social networking site. \({ }^{7}\) Find and interpret a \(95 \%\) confidence interval for each of the following proportions: (a) Proportion of US adults who use the Internet regularly. (b) Proportion of US adult Internet users who use a social networking site. (c) Proportion of all US adults who use a social networking site. Use the confidence interval to estimate whether it is plausible that \(50 \%\) of all US adults use a social networking site.

Short answer

The 95% confidence intervals would be: (a) for the proportion of US adults who use the Internet regularly, (b) for the proportion of US adult Internet users who use a social networking site, and (c) for the proportion of all US adults who use a social networking site. If for (c) the obtained interval contains 0.5, it is plausible that 50% of all US adults use a social networking site.

Step-by-step solution

Calculation of proportions

We need to calculate three different proportions. For (a), the proportion of US adults who use the Internet regularly, divide the number of regular Internet users by the total number of adults surveyed: 1787/2255. For (b), the proportion of US adult Internet users who use a social networking site, divide the number of social networking site users by the number of Internet users: 1054/1787. For (c), the proportion of all US adults who use a social networking site, divide the number of social networking site users by the total number of adults surveyed: 1054/2255.

Calculation of confidence intervals

Once we have the proportion (p), we can calculate a 95% confidence interval using this formula: \(p \pm z \sqrt{\frac{p(1-p)}{n}}\) where \(z\) is the z value that corresponds to a 95% level of confidence (approximately 1.96) and \(n\) is the total sample size. Apply this formula for each of the proportions from Step 1.

Interpretation of confidence intervals

The resulting intervals represent the range of proportions in which we can say with 95% confidence that the true proportion lies. Analyze each of the obtained intervals separately. For (c), compare the obtained interval with 0.5 (or 50%) to see if it is plausible that 50% of all US adults use a social networking site.

Key concepts

Proportion Calculation

Understanding proportion calculation is foundational when analyzing survey data. Proportion essentially represents a part of the whole, expressed as a fraction or percentage. For instance, in the survey mentioned, the proportion of US adults using the Internet regularly is calculated by dividing the number of regular Internet users (1787) by the total number of surveyed adults (2255). This yields a proportion of \(\frac{1787}{2255}\).

The same principle applies when calculating the proportion of US adult Internet users who use social networking sites. Here, we take the number of social networking site users (1054) and divide it by the number of Internet users (1787), resulting in \(\frac{1054}{1787}\). Lastly, to find the proportion of all US adults who use a social networking site, we divide the number of social networking site users (1054) by the total number of adults surveyed (2255), resulting in \(\frac{1054}{2255}\).

Proportions become particularly powerful when comparing different segments of a population or analyzing trends over time, and they serve as the basis for more complex statistical analyses, such as the construction of confidence intervals.

Survey Sampling

Survey sampling is a technique used to gather data from a subset of a population to make inferences about the whole. It's crucial to select a sample that accurately represents the population to ensure the results are generalizable. In our example survey, 2255 US adults were randomly selected. This random selection is vital because it helps to avoid sample bias, ensuring that every individual has an equal chance of being chosen and the sample's results are reflective of the entire population.

Key considerations in survey sampling include sample size and sampling method. Larger samples generally provide more reliable data, but practical considerations like cost and time also play a role in determining the optimal sample size. Different sampling methods (simple random, stratified, cluster, systematic) are utilized based on the research objectives and population structure.

It's also important to note that sample selection affects the precision of estimates and the width of confidence intervals: the larger the sample, typically, the narrower the confidence interval, indicating greater precision of the estimate.

Interpretation of Confidence Intervals

Interpreting confidence intervals is a critical skill in statistical analysis. A confidence interval provides a range within which we expect the true population parameter (in this case, a proportion) to lie, with a certain level of confidence. The 95% confidence interval used in our example survey implies that if we were to take 100 different samples and compute an interval from each sample, we would expect about 95 of those intervals to contain the true proportion.

In the context of the survey, after calculating the confidence interval for each of the three proportions, we can interpret these intervals to better understand the uncertainty around our estimates. For example, if the calculated confidence interval for the proportion of all US adults using a social networking site does not include 0.5 (50%), we can say with 95% confidence that it is not plausible that 50% of all US adults use such sites. This interpretation helps guide conclusions and decisions based on the survey data. Understanding confidence intervals thus equips us to handle statistical results more effectively and appreciate the role of chance and variability in sampled data.