Consider these sample means for the number of weekend hours spent studying for a Monday exam.
| Sample | Sample Mean |
| :----- | :---------- |
| 1 | 8 |
| 2 | 5 |
| 3 | 6 |
| 4 | 4 |
Which statements are true? Check all that apply.
A. The variation in the sample means is less than 5.
B. The variation in the sample means makes population predictions invalid.
C. The actual population mean most likely will be between 4 and 8.
D. The sample means can be used to make predictions about the population mean.
E. The population mean is greater than 8.
Answer (1)
Calculate the range of the sample means: r an g e = ma x − min = 8 − 4 = 4 .
Evaluate the truthfulness of each statement based on the calculated range and statistical principles.
The true statements are: 'The variation in the sample means is less than 5', 'The actual population mean most likely will be between 4 and 8', and 'The sample means can be used to make predictions about the population mean'.
The final answer is: \boxed{True, False, True, True, False}.
Explanation
Analyze the problem We are given four sample means: 8, 5, 6, and 4. We need to determine which statements about these sample means and the population mean are true.
Calculate the range First, let's find the range of the sample means. The range is the difference between the maximum and minimum values. In this case, the maximum value is 8 and the minimum value is 4. So, the range is 8 − 4 = 4 .
Evaluate each statement Now, let's evaluate each statement:
'The variation in the sample means is less than 5.' Since the range is 4, which is less than 5, this statement is true.
'The variation in the sample means makes population predictions invalid.' This statement is false. Sample means can be used to make predictions about the population mean, although there will be some uncertainty.
'The actual population mean most likely will be between 4 and 8.' This statement is true. The population mean is likely to fall within the range of the sample means.
'The sample means can be used to make predictions about the population mean.' This statement is true. Sample means are often used to estimate the population mean.
'The population mean is greater than 8.' This statement is uncertain. While it's possible, we cannot definitively say that the population mean is greater than 8 based on the given sample means.
Examples
Understanding sample means helps in many real-world scenarios. For example, if you're a quality control manager in a factory, you might take several samples of a product's weight. The sample means can help you estimate the average weight of all products made in the factory. If the sample means are close together, you can be more confident that the average weight of all products is close to the sample means. This helps ensure that the products meet the required specifications.
