Question

Consider taking a random sample of size 20 from the population of students at a certain college and measuring the diastolic blood pressure each of the 20 students. In the context of this setting, explain what is meant by the sampling distribution of the sample mean.

Short answer

The sampling distribution of the sample mean is the distribution of sample means over all possible samples of size 20 from the college student population. It shows how the mean diastolic blood pressure of a sample of 20 students would vary from one sample to another.

Step-by-step solution

Understanding the Concept of a Sampling Distribution

The sampling distribution of the sample mean refers to the probability distribution that specifies how the sample mean is distributed if we were to take many samples of the same size from the same population. It's a theoretical distribution of the means of all possible random samples of a given size drawn from a population.

Significance of Sample Size

In this case, the sample size is 20, which means for each sample from the population of college students, we measure the diastolic blood pressure of 20 randomly chosen students. We are considering the average diastolic blood pressure of these 20 students as a single observation from the sampling distribution.

Key Properties of the Sampling Distribution

The mean of the sampling distribution is equal to the mean of the population. The standard deviation of the sampling distribution, also known as the standard error, is the population standard deviation divided by the square root of the sample size (provided the population size is much larger than the sample size). This standard error measures the variability of the sample mean from sample to sample.

Key concepts

Sample Size

The term sample size refers to the number of individual data points collected in a single sample. It's a crucial factor in any statistical study as it affects the accuracy of the findings. For instance, a larger sample size generally leads to more precise estimates of the population characteristics, such as the mean or standard deviation. This is because as you increase the sample size, the sample's variability tends to decrease, resulting in a more precise estimate of the population parameter. In the context of diastolic blood pressure measurements among college students, a sample size of 20 means that 20 individuals are randomly selected for the study, and their blood pressure readings contribute to the calculation of an average that represents this sample.

Understanding the impact of sample size is vital, as it influences the standard error and the reliability of conclusions drawn from the data. When planning a study, researchers will often conduct a power analysis to determine the optimal sample size that balances practical constraints with the need for reliable and valid results.

Diastolic Blood Pressure

Diastolic blood pressure, one of the two main components measured when taking blood pressure readings, indicates the pressure in the arteries when the heart rests between beats. The American Heart Association considers a normal diastolic reading to be between 60 and 80 mmHg. It's essential to understand that diastolic blood pressure can vary among individuals and is influenced by various factors such as age, physical activity, and health conditions.

In a study involving college students, measuring the diastolic blood pressure provides information about their cardiovascular health. When researchers collect diastolic blood pressure data from a sample of students, they can use statistical methods to estimate the average blood pressure for the larger population of college students. This average derived from the sample is then a point of interest in understanding the overall health profile of the student body.

Standard Error

The standard error is a statistical term that quantifies the variability or precision of a sample statistic, such as a sample mean. It's crucial because it tells us how far off we might expect our sample estimate to be from the true population parameter if we were to repeat the study multiple times. The formula to calculate the standard error for the mean is \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population's standard deviation and \( n \) is the sample size. A smaller standard error indicates a more precise estimate of the population mean.

In the context of our example measuring diastolic blood pressure, if the population standard deviation of blood pressure is known or estimated, the standard error can be computed to understand the variability of the sample mean. Understanding standard error is essential for making inferences about the population, such as constructing confidence intervals and hypothesis testing.

Probability Distribution

A probability distribution is a mathematical function that describes the likelihood of different outcomes in an experiment or process. In statistics, the term often refers to the distribution of a sample statistic, such as the sample mean. The distribution shows us how probable different values of the sample mean are if we repeatedly take random samples from the population.

The shape and spread of the probability distribution for the sample mean depend on factors such as the population distribution, the sample size, and the variability within the data. In our exercise example, the sampling distribution of the sample mean would show us the spread of average diastolic blood pressure readings we might see if we were to take many samples of 20 students from the college population. This distribution is key to making inferences about the population mean and assessing the likelihood of observing a sample mean within a certain range.

Population Mean

The population mean is a parameter that represents the average value of a characteristic for an entire population. In the study of diastolic blood pressure, it would be the average pressure reading across all the college students, if we could measure every single one of them. However, since measuring every individual isn't feasible, we use a sample to estimate this population mean.

The sampling distribution of the sample mean is centered around the population mean, meaning that the most probable sample mean is the population mean itself. Understanding the population mean is vital because it serves as a benchmark for comparing sample statistics and determining whether a sample mean is significantly different from what is expected for the population. Collecting data and calculating the mean from a sample allows researchers to make educated guesses about the population mean, guiding them towards insights about the broader population's health or behavior.