The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends morethan a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy isadvertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of $115 and a standarddeviation of $19. If the management wants to give free gifts to at most 10.38% of the customers, what should the amount be abovewhich a customer would receive a free gift?Round your answer to two decimal places.A customer should spend at least $ito receive a free gift.

Question

28-10-2022
Mathematics

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The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends morethan a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy isadvertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of $115 and a standarddeviation of $19. If the management wants to give free gifts to at most 10.38% of the customers, what should the amount be abovewhich a customer would receive a free gift?Round your answer to two decimal places.A customer should spend at least $ito receive a free gift.

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NieveG660961 NieveG660961 · Answer 1 · 2022-10-28T16:11:45+02:00

The Z-score formula states that

[tex]Z=\frac{x-\mu}{\sigma}[/tex]

Where x is the value of the variable, mu is the mean of the distribution, and sigma is the standard deviation.

The value of Z defines an area below the curve that represents a certain percentage of the total data (100%), in a diagram

Therefore, we need to find a value of Z such that it covers 100%-10.38%=89.62% of the total data. This value can be found using a Z-score table of cumulative probability.

The Z-value that corresponds to 89.62% is Z=1.26

Then, solving the equation for x,

[tex]\begin{gathered} Z=1.26,\mu=115,\sigma=19 \\ \Rightarrow x=Z\sigma+\mu \\ \Rightarrow x=1.26\cdot19+115=138.94 \\ \Rightarrow x=138.94 \end{gathered}[/tex]

The answer is $138.94