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Answer :
Final answer:
Sample means often differ from the population mean due to sampling variability, with larger sample sizes reducing this variability. Paired samples are used in hypothesis testing to determine if the differences between them are significant.
Explanation:
Most of the sample means differ somewhat from the population mean due to sampling variability. This difference is a natural consequence of the random variation inherent in any data set. When multiple random samples are taken from the same population, the sample means will vary because each sample may contain different individuals or objects with various characteristics.
The more the sample means differ from one another, the greater the sampling variability. Sampling variability is influenced by the size of the sample; larger samples tend to produce means closer to the actual population mean, as they reduce the impact of outliers or individual variability.
In the context of hypothesis testing, researchers may use matched or paired samples to calculate differences and form another sample used for testing. When the sample sizes are sufficiently large, or the paired differences come from a normally distributed population, the sample mean of differences will approximate a normal distribution. This normality allows for the use of statistical tests like the Student's-t test to determine if there are significant differences between paired samples.
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