Question

In a survey taken in 1985, of the 21 households surveyed in Middletown, Ohio, the proportion of households having a personal computer was 1196. The U.S. government census data for 2005 suggested that 38% of households in the United States had a personal computer.

Which statement(s) below support the large discrepancy between the sample proportion of households having a personal computer in 1985 and the population proportion of households having a personal computer in 2005?
I. The sample size was too small.
II. The sample was not truly random.
III. The data from 1985 to 2005 have changed.
IV. The sample statistic is equal to the population parameter.
a) I and II only
b) II and III only
c) I, II, and IV only
d) III and IV only

Answer 1

The discrepancy between the sample proportion in the 1985 survey and the population proportion in 2005 can be explained by a small, potentially non-random sample size, and the expected significant changes in technology adoption over a 20-year period.

Step-by-step explanation:

The discrepancy between the sample proportion of households having a personal computer in 1985 and the population proportion in 2005 can be attributed to several factors. Firstly, the sample size of 21 households is indeed too small (option I) to be representative of the general population, especially for making comparisons over a 20-year period. This small sample size increases the margin of error and decreases the reliability of the results. Secondly, if the sample was not truly random (option II), this would introduce bias, rendering the results unreflective of the overall population.

However, the claim that 1196 (over 100%) of the households in the 1985 survey had a personal computer must be a typo or error since proportions cannot exceed 100%. Ignoring the likely misprint, the increase in technology adoption by 2005 is a natural progression, indicating that the data from 1985 to 2005 are expected to show significant changes (option III) due to advancements in technology and the decrease in computer costs. Lastly, option IV is incorrect because the sample statistic cannot be equal to the population parameter unless the entire population was surveyed.

Taking into consideration these points, options I, II, and III provide the best explanation for the large discrepancy. The correct answer should capture the small and potentially biased nature of the initial sample, alongside the extensive technological advancement and increased adoption of personal computers from 1985 to 2005.