Question

a. Use the given information to estimate the proportion of college students who use the Internet more than 3 hours per day. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.Most American college students make use of the Internet for both academic and social purposes. What proportion of students use it for more than 3 hours a day? The authors of the paper "U.S. College Students" Internet Use: Race, Gender and Digital Divides" (Journal of Computer-Mediated Communication [2009]: 244-264) describe a survey of 7,421 students at 40 colleges and universities. The sample was selected to reflect general demographics of U.S. college students. Of the students surveyed, 2,998 reported Internet use of more than 3 hours per day.

Short answer

The proportion of college students using the internet for more than three hours daily is \( \frac{2998}{7421}\). The calculated margin of error can be interpreted as the range within which the true proportion of such students is likely to fall, with chosen level of confidence.

Step-by-step solution

Calculation of Proportion

The proportion of students using the Internet for more than three hours daily can be estimated by dividing the number of such students by the total surveyed. Hence, the proportion is calculated as: \( p = \frac{2998}{7421}\)

Verification of conditions for Margin of Error

For the margin of error formula to apply, the sample proportion must be roughly normal. This is verified if \(np \geq 10\), \(n(1-p) \geq 10\) where n is the sample size and p is the proportion. Plug in the values to verify.

Calculation of Margin of Error

The margin of error, E, is calculated as \( E = Z\sqrt{\frac{p(1-p)}{n}} \) where Z is the confidence level expressed in Z-score (usually, Z=1.96 for 95% confidence), p is the proportion, and n is the sample size. Substitute the values to calculate the margin of error.

Interpretation of Margin of Error

The margin of error indicates the range within the true proportion is likely to fall with given confidence level. If the calculated margin of error is E, then the confidence interval is (p-E, p+E). It means that we are confident that the true proportion of students using the internet more than three hours daily, lies in this interval.

Key concepts

Proportion Estimation

In statistics, proportion estimation is a fundamental concept used to approximate the part of a population that has a certain characteristic based on sample data. For instance, if we want to estimate the proportion of college students who use the Internet for more than three hours per day, we start by taking a sample that is representative of the population, in this case, a survey involving 7,421 students across different colleges.

To estimate the proportion (\( p \)), the number of students who have the characteristic of interest (in this case, Internet use exceeding three hours) is divided by the total number of students surveyed. Mathematically, it is expressed as: \( p = \frac{\text{number of students with the characteristic}}{\text{total number in the sample}} \)

In the sample provided, 2,998 out of 7,421 students reported excessive Internet use, so the estimate of the proportion is \( p = \frac{2998}{7421} \). It's essential to remember that this figure is an estimate of the actual proportion in the larger population of U.S. college students. Ensuring that the sample is representative of the population helps improve the accuracy of this estimation.

Sample Size

Sample size refers to the number of observations or individuals included in a study and holds paramount significance in statistical analyses. It is denoted by the symbol \( n \). The accuracy of an estimate from a sample, like the proportion estimation, depends largely on the size of the sample.

To ensure reliability in the margin of error calculations, specific conditions related to sample size must be met. These conditions often include the requirements that \( np \geq 10 \) and \( n(1-p) \geq 10 \), where \( p \) is the sample proportion. These inequalities ensure that the sample is large enough for the approximation to the normal distribution to be reasonable, allowing for further inferences about the population to be made with confidence.

In this problem's context, verifying these conditions with the values provided (\( n = 7421 \) and \( p = \frac{2998}{7421} \) ) ensures that the sample is adequate for the margin of error formula to be applicable. Large sample sizes typically result in a smaller margin of error and a narrower confidence interval, implying a more precise estimate.

Confidence Interval

A confidence interval gives a range of values for a population parameter, such as a proportion, estimated from the sample data. It is usually presented as the sample statistic plus or minus a margin of error. The confidence interval is a way to express how certain we are about an estimate. It is intrinsically connected to the concept of a confidence level, which typically might be 90%, 95%, or 99%.

The margin of error (\( E \) ) is computed using the formula \( E = Z\sqrt{\frac{p(1-p)}{n}} \) where \( Z \) corresponds to the Z-score related to the confidence level, \( p \) is the proportion estimated from the sample, and \( n \) is the sample size. For example, a 95% confidence level has a Z-score of about 1.96. After calculating the margin of error, the confidence interval can be constructed.

If the margin of error for the estimated proportion of students using the Internet more than three hours a day is found to be \( E \) based on a 95% confidence level, the confidence interval would be given by (p-E, p+E). This interval tells us that we can be 95% confident that the true proportion of students with this level of Internet usage lies within this range, thus providing a level of precision to our estimate.