A$/$B Testing, the test ~ The Peaches Skin Care company is developing a new marketing campaign for one of its skin care products. Because the skin care market is crowded and complex and because advertising is a large expense, the company decides to use a procedure called A$/$B testing on its new advertisement.

For a pilot study, the marketing team at the company obtains a group of $410$ potential customers for the product. They randomly assign $272$ potential customers to watch the advertisement for the product $($condition A$)$ and $138$ potential customers to watch a public service announcement $($PSA$)$ of the same length and size as the advertisement $($condition B$).$ They then record whether the potential customer purchased the product or not.

The results from this pilot study are shown in the table below.

Condition Purchased Did not purchase Total

A: Advertisement $25247272$

B: PSA $5133138$

Total $30380410$

$1.$ Calculate the observed difference in the proportion of study participants in Condition A and Condition B who purchased the product.

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