Question

How much sleep do you get on a typical school night? A group of 10 college students were asked to report the number of hours that they slept on the previous night with the following results: $$ \begin{array}{llllllllll} 7, & 6, & 7.25, & 7, & 8.5, & 5, & 8, & 7, & 6.75, & 6 \end{array} $$ a. Find a \(99 \%\) confidence interval for the average number of hours that college students sleep. b. What assumptions are required in order for this confidence interval to be valid?

Short answer

Based on the given data, the 99% confidence interval for the average number of sleep hours for college students is between 5.53 and 8.17 hours. The underlying assumptions required for the confidence interval to be valid include random sampling of college students, either a normal distribution of the sample data or a large enough sample size to ensure normal distribution through the Central Limit Theorem, and the use of the sample standard deviation since the population standard deviation is unknown.

Step-by-step solution

Calculate the sample mean and standard deviation

First, we need to find the sample mean (\(\bar{x}\)) and standard deviation (s) of the given sleep hours data. The sample mean can be found by summing up the data points and dividing by the number of data points (n). \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\) For standard deviation, we will use the following formula for a sample: \(s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}\) Using the given data: \(\bar{x} = \frac{7+6+7.25+7+8.5+5+8+7+6.75+6}{10} = \frac{68.50}{10} = 6.85\) \(s = \sqrt{\frac{(0.85^2) + (-0.85^2) + (0.4^2) + (0.85^2) + (1.65^2) + (-1.85^2) + (1.15^2) + (0.85^2) + (-0.1^2) + (-0.85^2)}{9}} = 1.27\)

Apply t-distribution to compute the confidence interval

Now, we will use a t-distribution with a 99% confidence level and 9 degrees of freedom (which is n-1 = 10-1 = 9) to find the confidence interval. The formula for the confidence interval is: \(\bar{x} \pm t_{\alpha/{2}} \frac{s}{\sqrt{n}}\) By looking up a t-table, we find that \(t_{\alpha/{2}} = 3.25\) for \(99\%\) confidence and 9 degrees of freedom. So the confidence interval is: \(6.85 \pm 3.25\frac{1.27}{\sqrt{10}}\) \(6.85 \pm 1.32\) Finally, the 99% confidence interval is: \((5.53, 8.17)\)

Assumptions required for the confidence interval

In order for this confidence interval to be valid, the following assumptions must be met: 1. The sample data is randomly selected. This means that each college student has an equal likelihood of being included in the sample. 2. The sample should have a normal distribution or come from an underlying population that is normally distributed. If the sample size is large enough, the Central Limit Theorem can ensure that the sample mean will be approximately normally distributed. 3. The population standard deviation is not known, so we use the sample standard deviation (which is why we use a t-distribution instead of a z-distribution). In conclusion, we found the 99% confidence interval for the average number of hours that college students sleep to be between 5.53 and 8.17 hours. For this confidence interval to be valid, the above-mentioned assumptions must hold true.

Key concepts

t-distribution

When dealing with smaller sample sizes, typically less than 30, we utilize the t-distribution instead of the normal distribution to construct confidence intervals. The t-distribution accounts for the additional uncertainty we have when estimating the population standard deviation using a smaller sample size.
This distribution has fatter tails compared to the normal distribution, which allows for more variability in the data. It is particularly useful when we do not know the population standard deviation, but rather use the sample's standard deviation.
The t-distribution approach requires determining the degrees of freedom, calculated as the sample size minus one ( -1). In our problem, the sample size was 10, which means 9 degrees of freedom were used to find the corresponding t-value for the 99% confidence interval. This t-value helps adjust our expectations and provides a more accurate range for our confidence interval.

sample mean

The sample mean is a critical statistic that represents the average of the data points in a sample. It provides a central value for the data and is often used as an estimate for the population mean, especially when we’re dealing with smaller datasets.
In the sleep study, the sample mean (\bar{x}) was calculated by adding all the reported hours of sleep by the students and then dividing by the number of students. Mathematically, it was \(\frac{7+6+7.25+7+8.5+5+8+7+6.75+6}{10} \), resulting in a mean of 6.85 hours.
Having the mean is crucial, as it becomes a key component in further statistical calculations, such as constructing a confidence interval, thereby providing insight into the average behavior of the entire population based on the sample.

standard deviation

Standard deviation is a measure of how spread out the numbers in a dataset are. It offers insight into the variability or dispersion of the sample data points around the mean.
In statistical analysis, a smaller standard deviation indicates that the data points tend to be closer to the mean, while a larger standard deviation suggests more spread out data. In our sleep study, the standard deviation was calculated to be 1.27 hours, indicating some variation in the sleep patterns of the students.
The formula used is specifically for sample data, which divides the squared deviations from the mean by \(n-1\) (where n\ is the sample size), making it more reliable for smaller samples. This adjusted calculation allows it to act as an unbiased estimate of the population standard deviation.

random sampling

Random sampling is an essential assumption when performing statistical analyses and constructing confidence intervals. It ensures that every individual in the population has an equal chance of being selected for the sample, which is critical for maintaining the objectivity and reliability of the results obtained.
If the sampling is not random, there's a risk of introducing bias, potentially leading to incorrect conclusions. In the context of our sleep study, it was assumed that the group of college students was chosen randomly, meaning no particular individual had a better chance of being sampled than another.
Random sampling helps in reflecting the true characteristics of the entire population in the sample, thereby allowing statisticians to make more valid generalizations and ultimately influencing the accuracy and trustworthiness of the confidence interval.