Question

javier and ercilla are supervisors at a shopping mall. each was given the task of estimating the mean distance that shoppers live from the mall. they each randomly surveyed 100 shoppers. the samples yielded the following information.

	javier	ercilla
$s$	4.0 miles	7.0 miles

(a) how can you determine which survey was correct?

• ercilla’s survey was correct because the standard deviation is greater than the mean distance shoppers live from the mall.
• javier’s survey was correct because the standard deviation is only 2.0 miles away from the mean distance shoppers live from the mall.
• ercilla’s survey was correct because the standard deviation is only 1.0 mile away from the mean distance shoppers live from the mall.
• javier’s survey was correct because the standard deviation is less than the mean distance shoppers live from the mall.
• there is no way to determine from these numbers which survey was correct.

(b) explain what the difference in the results of the surveys implies about the data.

• the difference in the results shows that javier’s sample had a smaller standard deviation than ercilla’s sample. this shows that there is not any variability in the distances that people live from the mall.
• the difference in the results shows that javier’s sample had a larger standard deviation than ercilla’s sample. this shows that, although the mean distance appears to be around 6 miles, there is probably some variability in the distances that people live from the mall.
• the difference in the results shows that javier’s sample had a smaller standard deviation than ercilla’s sample. this shows that, although the mean distance appears to be around 6 miles, there is probably some variability in the distances that people live from the mall.
• the difference in the results shows that javier’s sample had a larger standard deviation than ercilla’s sample. this shows that there is not any variability in the distances that people live from the mall.

(c) if the two histograms depict the distribution of values for each supervisor, which one depicts ercilla’s sample? how do you know?
two histograms are shown, (a) has a peak at 6 with varying bar heights, (b) has uniform bar heights

• histogram (a) depicts ercilla’s sample because it appears to show a larger range of values.

Explanation:

Part (a)

To determine which survey is correct, we analyze the mean ($\bar{x}$) and standard deviation ($s$). The mean distance is 6.0 miles for both. Standard deviation measures data spread. There's no rule that standard deviation must be less than or greater than the mean to determine correctness. The numbers (mean and standard deviation) alone don't indicate which survey is correct as both are valid samples with different spreads.

Javier’s standard deviation ($s = 4.0$) is smaller than Ercilla’s ($s = 7.0$). A smaller standard deviation means less variability in the data, and a larger one means more variability. The mean is 6.0 for both, so the difference in standard deviations implies variability: Javier’s sample has less spread (smaller $s$), Ercilla’s has more (larger $s$). The correct option explains that Javier’s smaller $s$ shows some (less) variability around the mean.

Ercilla’s standard deviation ($s = 7.0$) is larger than Javier’s ($s = 4.0$), meaning her data has a larger spread (range of values). Histogram (a) shows a larger range (more spread out) compared to (b) (more uniform or less spread). So the histogram with the larger range (more spread) depicts Ercilla’s sample.

Answer:

There is no way to determine from these numbers which survey was correct.

Part (b)