"Sleepy Suppose you want to study the number of hours of sleep you get each evening. To do so, you look at the calendar and randomly select 10 days out of the next 300 days and record the number of hours you sleep.


d. Is the mean number of hours computed in part (c) a random variable? Why? If it is a random variable, what is the source of variation"

Verified step by step guidance

1. Understand what a random variable is: it is a numerical outcome of a random phenomenon or experiment.
2. Recognize that in this problem, the mean number of hours of sleep is calculated from a sample of 10 randomly selected days out of 300 days.
3. Since the 10 days are randomly selected, the sample mean depends on which days are chosen, making it a random quantity that can vary from sample to sample.
4. Therefore, the mean number of hours computed is a random variable because it varies depending on the random selection of days.
5. The source of variation is the randomness in the selection of the 10 days from the 300 days, which causes the sample mean to change with different samples.

Key Concepts

Random Variable

A random variable is a numerical outcome of a random phenomenon or experiment. It assigns a number to each possible outcome in a sample space, and its value can vary due to chance. In this context, the mean number of hours of sleep calculated from a random sample is a random variable because it depends on which days are selected.

Sampling and Sample Mean

Sampling involves selecting a subset of data from a larger population to make inferences. The sample mean is the average of observations in the sample and serves as an estimate of the population mean. Since the sample is randomly chosen, the sample mean varies with different samples, making it a random variable.

Source of Variation in Sampling

The source of variation in the sample mean arises from the randomness in selecting different samples. Each random sample may include different days with varying sleep hours, causing the sample mean to fluctuate. This variability is known as sampling variability and is fundamental in statistical inference.