Question

javier and ercilia 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	ercilia
$s$	4.0 miles	7.0 miles

(a) how can you determine which survey was correct?
\bigcirc javier’s survey was correct because the standard deviation is only 2.0 miles away from the mean distance shoppers live from the mall.
\bigcirc ercilia’s survey was correct because the standard deviation is only 1.0 mile away from the mean distance shoppers live from the mall.
\bigcirc javier’s survey was correct because the standard deviation is less than the mean distance shoppers live from the mall.
\bigcirc ercilia’s survey was correct because the standard deviation is greater than the mean distance shoppers live from the mall.
\bigcirc 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.
\bigcirc the difference in the results shows that javier’s sample had a smaller standard deviation than ercilia’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.
\bigcirc the difference in the results shows that javier’s sample had a smaller standard deviation than ercilia’s sample. this shows that there is not any variability in the distances that people live from the mall.
\bigcirc the difference in the results shows that javier’s sample had a larger standard deviation than ercilia’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.
\bigcirc the difference in the results shows that javier’s sample had a larger standard deviation than ercilia’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 ercilia’s sample? how do you know?
two histograms: (a) has a peak around 6, (b) is more uniform
\bigcirc histogram (a) depicts ercilia’s sample because it appears to show a larger range of values.
\bigcirc histogram (b) depicts ercilia’s sample because it appears to show a larger range of values.

Explanation:

Part (a)

To determine which survey is correct, we analyze the options. The mean (\(\bar{x}\)) for both is 6.0 miles, but standard deviation (\(s\)): Javier has 4.0, Ercilia has 7.0. Standard deviation measures variability, but there's no rule that \(s\) must be closer or follow a specific relation to \(\bar{x}\) to determine correctness. So we can't tell from these numbers.

Javier’s \(s = 4.0\), Ercilia’s \(s = 7.0\). So Javier’s sample has smaller \(s\) (less variability) than Ercilia’s (more variability). The correct option explains that smaller \(s\) for Javier, and there is still variability (since \(s>0\)) around the mean (6 miles).

Ercilia has a larger standard deviation (\(s = 7.0\)), meaning her data has a larger range (more spread out). Histogram (b) looks more spread out (uniform, wider range) than (a) (peaked at 6). So Histogram (b) depicts Ercilia’s sample as it shows a larger range.

Answer:

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

Part (b)