Question

On a nationally standardized reading comprehension test, the norm for the population mean was designed to yield a scaled score value of 50 for the population of fifth-grade students. A random sample of 10 elementary school children was tested, yielding scores of 48,42,55,35,50,47,45,39.and 42. Estimate whether this sample might be representative of the population for whom the test was designed.

Answer 1

The sample mean for the fifth-graders' reading test is lower than the population mean, suggesting the sample might not be representative. Further statistical analysis, such as hypothesis testing or constructing a confidence interval, would be needed to make a more definitive assessment.

Step-by-step explanation:

To determine whether the sample is representative of the fifth-grade student population, we can compute the sample mean and compare it to the known population mean, which is a scaled score of 50. The sample mean is calculated by summing all the scores and dividing by the number of scores.

Sample mean = (48 + 42 + 55 + 35 + 50 + 47 + 45 + 39 + 42) / 9 ≈ 44.89

As the sample mean of 44.89 is lower than the population mean of 50, it might indicate that the sample is not perfectly representative of the population. However, with such a small sample size (n=9), the result could be due to randomness or sampling error. To estimate whether the sample is actually from the population, a hypothesis test could be performed, or a confidence interval could be computed to see if the population mean falls within that interval.

Confidence intervals provide a range of values that are likely to contain the population mean a certain percentage of the time (e.g., 95% or 90%). For example, if we say we estimate with 95 percent confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98, this means that if we were to take many samples and compute a 95% confidence interval for each sample, about 95% of these intervals would contain the true population mean.