Cash needed to buy two normally priced jerseys

Question

Two football jerseys have random, independent prices: the first is normal with mean $80 and standard deviation $15, the second normal with mean $70 and standard deviation $20. Gabe wants a $97.5\%$ chance of being able to afford both. How many dollars should he carry? Take the standard normal's $95$ th percentile to be $1.65$ and its $97.5$ th percentile to be $1.96$ .

Short answer

$199. The total price is normal with mean $150$ and variance $15^2+20^2=625$ , i.e. standard deviation $25$ . Its $97.5\%$ quantile is $150+1.96\cdot25=199$ .

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