Question
If you take 3 different random samples of 20 students each from a grade of 200 students, would you expect the mean height to be exactly the same for each sample?
Explain your reasoning.
Explain your reasoning.
Answers
GPT-4o mini
No, you would not expect the mean height of each of the three random samples of 20 students to be exactly the same. Here are a few reasons for this:
1. **Sampling Variability**: Each random sample of 20 students is taken from the same population of 200 students, but due to the inherent variability in the population, different samples can include different individuals. This means that the heights (or any measurement) of the students in each sample will vary, leading to differences in the calculated sample means.
2. **Random Selection**: The process of random sampling means that each student has an equal chance of being selected for each sample. Because of this randomness, even if the overall population mean height is a specific value, the means of the different samples can fluctuate around this population mean due to the specific individuals chosen in each sample.
3. **Law of Large Numbers**: While the mean height of a large number of samples will tend to be close to the population mean when averaged, each individual sample itself will likely deviate from the population mean. As the size of samples increases, the sample means will converge to the population mean, but for smaller samples like 20 students, variability is more pronounced.
4. **Statistical Distribution**: The means of the samples will follow a distribution (specifically, the sampling distribution of the sample mean), which implies that there will be a range of possible mean heights for different samples taken from the same population.
In conclusion, while all three samples are drawn from the same population, factors such as sampling variability and randomness ensure that the means calculated from each sample are unlikely to be exactly the same. They will likely cluster around the population mean but with some variation.
1. **Sampling Variability**: Each random sample of 20 students is taken from the same population of 200 students, but due to the inherent variability in the population, different samples can include different individuals. This means that the heights (or any measurement) of the students in each sample will vary, leading to differences in the calculated sample means.
2. **Random Selection**: The process of random sampling means that each student has an equal chance of being selected for each sample. Because of this randomness, even if the overall population mean height is a specific value, the means of the different samples can fluctuate around this population mean due to the specific individuals chosen in each sample.
3. **Law of Large Numbers**: While the mean height of a large number of samples will tend to be close to the population mean when averaged, each individual sample itself will likely deviate from the population mean. As the size of samples increases, the sample means will converge to the population mean, but for smaller samples like 20 students, variability is more pronounced.
4. **Statistical Distribution**: The means of the samples will follow a distribution (specifically, the sampling distribution of the sample mean), which implies that there will be a range of possible mean heights for different samples taken from the same population.
In conclusion, while all three samples are drawn from the same population, factors such as sampling variability and randomness ensure that the means calculated from each sample are unlikely to be exactly the same. They will likely cluster around the population mean but with some variation.