Questions: Which of the following statements are true? Select all that apply. The mean of Sample 2 underestimates the population mean. The mean of Sample 1 underestimates the population mean. The mean of Sample 2 overestimates the population mean. The mean of Sample 2 equals the population mean. The mean of Sample 1 overestimates the population mean. The mean of Sample 1 equals the population mean.

Solution Steps

To determine which statements about the means are true, we need to calculate the population mean and the means of the given samples. Then, we can compare the sample means to the population mean to see if they underestimate, overestimate, or equal the population mean.

1. Calculate the population mean using the scores of all students.
2. Calculate the mean of Sample 1.
3. Calculate the mean of Sample 2.
4. Compare the sample means to the population mean to determine which statements are true.

The population mean is calculated using the scores of all students: \[ \text{Population Mean} = \frac{97 + 81 + 78 + 55 + 98 + 84 + 93 + 87 + 61 + 98 + 100 + 92}{12} = 85.3333 \]

The mean of Sample 1 is calculated as follows: \[ \text{Mean of Sample 1} = \frac{97 + 81 + 78 + 55 + 98 + 84}{6} = 82.1667 \]

The mean of Sample 2 is calculated as follows: \[ \text{Mean of Sample 2} = \frac{93 + 87 + 61 + 98 + 100 + 92}{6} = 88.5000 \]

• The mean of Sample 1 (\(82.1667\)) is less than the population mean (\(85.3333\)), so it underestimates the population mean.
• The mean of Sample 2 (\(88.5000\)) is greater than the population mean (\(85.3333\)), so it overestimates the population mean.

Final Answer