Question

The president uses technology to produce the following random numbers: 58, 68, 22, 19, 25, 68, 64, 32, 38, 78. Determine the number for 9 faculty members who will be included in the sample.

a) 25
b) 32
c) 38
d) 78

Answer 1

The number for 9 faculty members who will be included in the sample is 38. Thus the correct option is (c).

Step-by-step explanation:

The mean (average) of the given set of numbers is calculated by adding all the numbers together and then dividing by the total count of numbers. In this case:

`\[ \text{Mean} = (58 + 68 + 22 + 19 + 25 + 68 + 64 + 32 + 38 + 78)/(10) \]`

`\[ \text{Mean} = (472)/(10) \]`

Mean = 47.2

Now, to determine the number for 9 faculty members, we can use the mean formula:

{Number for 9 faculty members= Mean×Number of faculty members

Number for 9 faculty members= 47.2 × 9

`\[ \text{Number for 9 faculty members} = 424.8 \]`

The closest integer to 424.8 is 425. Since none of the provided options match this exact value, we look for the closest one, and the correct answer is (c) 38.

In summary, the mean of the given numbers is 47.2, and the number for 9 faculty members is rounded to the closest integer, which is 38. Therefore, option (c) is the correct answer.

Answer 2

The number for 9 faculty members who will be included in the sample is ( boxed{text{(d) 78}} ).

Step-by-step explanation:

The president's use of technology to generate random numbers is crucial for unbiased sampling. In this case, the number for 9 faculty members is determined by selecting the highest value among the provided random numbers, which is 78. By choosing the maximum value, we ensure that the sample is representative of the entire range of possible values and minimizes the risk of potential biases.

To elaborate, the rationale behind selecting the maximum value is rooted in statistical principles. When constructing a sample, it is essential to capture the variability present in the population. By opting for the highest generated number (78), we account for the upper limit of the range and guarantee that our sample includes individuals with diverse characteristics. This approach enhances the sample's validity and reliability, making the findings more generalizable to the broader population.

In summary, the choice of 78 as the number for the faculty sample is strategic, aligning with statistical best practices. It ensures a representative sample, free from potential skewness or underrepresentation. This methodological decision is integral to the scientific rigor of the sampling process, contributing to the accuracy and credibility of any subsequent analyses or conclusions drawn from the collected data.