Since a sample is a subset of the population , the sample mean
a. is always smaller thant he mean of the population
b,. si always larger than the mean of the population
c. must be equal to the mean of the population
d. can be larger or smalller or equal to the mean of the population.
The Correct Answer and Explanation is :
Correct Answer:
d. The sample mean can be larger, smaller, or equal to the population mean.
Explanation:
A sample is a subset of a population, and the sample mean is an estimate of the population mean. However, due to randomness in sampling, the sample mean may not always be exactly equal to the population mean.
Understanding the Relationship Between Sample Mean and Population Mean
- Random Sampling and Variation
- A sample is taken randomly from a population. The specific values included in the sample depend on which individuals are selected.
- Some samples may contain values that are mostly lower than the population mean, while others may contain values that are mostly higher.
- Law of Large Numbers
- If we take many random samples and compute their means, these sample means will fluctuate.
- However, as the sample size increases, the sample mean tends to get closer to the population mean.
- Sampling Error
- The difference between the sample mean and the population mean is called the sampling error.
- This error occurs because the sample may not fully represent the entire population.
- Depending on the specific individuals included in the sample, the sample mean can be higher, lower, or equal to the population mean.
- Example
Suppose we have a population of 1,000 students, and the average height (population mean) is 165 cm.
- If we randomly select a sample of 50 students, the sample mean could be 164 cm, 167 cm, or even exactly 165 cm, depending on which students are selected.
Conclusion
Since the sample mean is based on a subset of the population, it can be larger, smaller, or equal to the population mean. Therefore, option (d) is correct.