A researcher wants to estimate the average distance that...

A researcher wants to estimate the average distance that students at a large community college live from campus. To find out, she surveys a simple random sample of students from the registrar's database. (a) Will the researcher's sample result be exactly the same as the true population mean? Explain. (b) Which would be more likely to get the researcher's sample result closer to the true population value: an SRS of 100 students or an SRS of 50 students?Explain.

A researcher wants to estimate the average distance that students at a large community college live from campus. To find out, she surveys a simple random sample of students from the registrar's database. (a) Will the researcher's sample result be exactly the same as the true population mean? Explain.
(b) Which would be more likely to get the researcher's sample result closer to the true population value:
an SRS of 100 students or an SRS of 50 students?Explain.

Key Concepts

Sampling Error

Sampling error refers to the inherent difference between a sample statistic (like the sample mean) and the corresponding population parameter (like the true mean) due to the random selection of a sample. Even though a simple random sample is unbiased, individual samples may yield different estimates because they capture only a subset of the entire population, leading to natural variability in the results.

Simple Random Sampling (SRS)

Simple random sampling is a method where each member of the population has an equal chance of being selected for the sample. This technique helps to reduce bias and ensures that the sample is representative of the population, making statistical inferences more valid even though the sample may not precisely match the population parameters due to random variation.

Sampling Distribution

The sampling distribution is the probability distribution of a given statistic (for example, the sample mean) based on repeated sampling from the same population. It is centered around the true population parameter, and its spread is determined by the standard error, which decreases as the sample size increases. This concept explains why individual samples might differ from the true value but, on average, are accurate representations of the population.

Effect of Sample Size

Increasing the sample size reduces the standard error of the sample mean, making the estimate more precise. In a larger simple random sample, the variability due to random sampling decreases, which increases the likelihood that the sample mean will be closer to the true population mean. Conversely, a smaller sample size tends to result in greater variability and less precision in the estimate.

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