Question

In the dataset StudentSurvey, 361 students recorded the number of hours of television they watched per week. The average is \(\bar{x}=6.504\) hours with a standard deviation of \(5.584 .\) Find a \(99 \%\) confidence interval for \(\mu\) and interpret the interval in context. In particular, be sure to indicate the population involved.

Short answer

The 99% confidence interval for the average number of hours all students watch television per week is between 5.75 hours and 7.258 hours.

Step-by-step solution

Determine the Confidence Level

First, recognize that a 99% confidence interval corresponds to \(\alpha = 0.01\). This leaves 0.005 in each tail of the Gaussian distribution. From the z-table or using a calculator, find the z-score for \(0.005\) which is approximately \(2.576\).

Calculate the Standard Error

Next, calculate the standard error by dividing the standard deviation by the square root of the number of observations. The standard error is \(SE = 5.584 / \sqrt{361} = 0.293\).

Form the Confidence Interval

Now, apply the formula for the confidence interval, which is \(\bar{x} \pm z * SE\). Plug in the numbers: \(6.504 \pm 2.576 * 0.293\). Calculating the numbers gives the range \(6.504 \pm 0.754\).

State the Confidence Interval

The 99% confidence interval for \(\mu\) is therefore (5.75 hours, 7.258 hours).

Interpret the Confidence Interval

In the context of the problem, this means we are 99% confident that the average number of hours all students watch television per week lies between 5.75 hours and 7.258 hours.

Key concepts

Standard Error

The standard error (SE) is a measure that describes how much the sample mean, denoted as \( \bar{x} \), is expected to differ from the true population mean, represented by \( \mu \). It's a key statistic used to gauge the precision of our sample mean estimate. In our exercise, it's vital to calculate the standard error to determine the level of confidence we can have in our interval estimate of the population mean.

To calculate the SE, we divide the standard deviation (\( \sigma \)) of the sample by the square root of the sample size (n), giving the formula: \( SE = \frac{\sigma}{\sqrt{n}} \). In the context of the provided dataset, with a standard deviation of 5.584 and 361 students, the formula then becomes \( SE = \frac{5.584}{\sqrt{361}} = 0.293 \). This small SE indicates a higher precision of our sample mean estimate, which, in turn, contributes to a narrower confidence interval, leading to a more accurate estimate of the population mean.

Z-Score

A z-score is a type of standard score that tells us how many standard deviations an element is from the mean. It's pivotal in statistics for standardizing different data sets to be comparable, and is commonly used in the calculation of confidence intervals. When constructing a confidence interval for the population mean, one such z-score corresponds to the chosen confidence level.

In our exercise, with a confidence level of 99%, the z-score marks the point on a standard normal distribution where 99% of the values lie between. There is 0.5% of the data on each side of the distribution that falls outside of this range — these are the tails. For a 99% confidence interval, we seek the z-score where 0.5% is in the tail, and we get approximately 2.576. This z-score is then used to multiply by the standard error in order to expand our sample mean to an interval that is likely to contain the population mean with a certain level of confidence.

Population Mean

The population mean \( \mu \) is a parameter that represents the average of a given quantity across the entire population. In statistical terms, 'population' refers to the full set of observations that can be made. It's important to differentiate between the population mean and the sample mean, which is the average calculated from data collected from a subset of the population.

In the context of the StudentSurvey exercise, while we calculate the sample mean (\( \bar{x} = 6.504 \) hours) of television watching time from 361 students, we aim to infer the population mean— which would be the average viewing time of all students. Confidence intervals are instrumental in this scenario. They provide an estimated range where the population mean is likely to fall. Understanding the population mean helps in researching behaviors, making policies, and drawing conclusions about the broader student community's television consumption habits.

Statistics

Statistics as a discipline involves collecting, analyzing, interpreting, presenting, and organizing data. It provides methods to perform these tasks efficiently and ways to make sense of numerical data in order to make informed decisions. Concepts such as standard error, z-scores, and confidence intervals all fall under the umbrella of statistics and are crucial for understanding variability and uncertainty inherent in data.

Through the application of statistical methods in the given exercise, we're able to estimate the average number of hours students spend watching television with a quantifiable level of certainty—expressed through the 99% confidence interval. Statistics not only aids in making such inferences but also equips us with the tools to communicate the reliability and precision of our estimates, which is particularly valuable in fields dealing with extensive data analysis.