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In the context of this exercise, a proportion refers to a mathematical representation of the part of the sample exhibiting a certain behavior or attribute. It is expressed as a fraction of the total sample or population size. Here, we aim to estimate the proportion of college students who check their phones for purposes other than checking the time at least twice a night. This is calculated using the formula \( \hat{p} = \frac{x}{n} \), where \( x \) represents the number of students showing the behavior (125), and \( n \) is the total number of surveyed students (734). Hence, the estimated proportion \( \hat{p} \) is approximately 0.17, indicating that 17% of students in the sample have this behavior.

Proportions are useful in statistics as they provide a ratio that makes it easier to understand how one part compares to the entire set. It simplifies comparisons and predictions about a population based on a sample.

A confidence interval is a range of values derived from sample data that is likely to contain the true value of an unknown population parameter. It provides a measure of certainty about our sample estimates. In this problem, we're constructing a 95% confidence interval for the proportion \( \hat{p} \) of students who check their phones frequently at night. To calculate this interval, we use the formula: \[ CI = \hat{p} \pm ME \]where \( ME \) is the margin of error.

The margin of error (ME) reflects the degree of uncertainty around our estimate. In this case, it's calculated as \( ME = Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( Z = 1.96 \) for a 95% confidence interval, \( \hat{p} \approx 0.17 \), and \( n = 734 \). The resulting margin of error is approximately 0.0333, which means our 95% confidence interval for the proportion is (0.1367, 0.2033).

This suggests that we are 95% confident that the actual proportion of all college students engaging in this behavior is between 13.67% and 20.33%. Confidence intervals are crucial in research as they provide a range around the estimate, which accounts for natural variations in the sample.

Random sampling is a fundamental concept in statistics that refers to selecting a subset of individuals from the population where each individual has an equal chance of being chosen. This approach is vital for making sure that the collected sample represents the entire population without bias, which in turn ensures that the results and inferences made about the population are reliable.

In this exercise, we assume the sample of 734 students is representative of the broader population of college students in the United States. Random sampling ensures that our sample estimates, like the proportion of students checking their phones at night, are unbiased and generalizable.

• Random sampling reduces sampling bias, which can skew data and lead to incorrect conclusions.
• It helps in creating a diverse sample that mirrors the broader population, making the results more valid across various subgroups.

By assuming random sampling, we confidently apply statistical formulas, such as those for margin of error and confidence intervals, with the assurance that our sample’s characteristics align well with those of the target population. This practice is central to drawing meaningful insights in statistical studies.