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Relatively basic question here ..

A Company wants to give gifts to the top 5% of customers. If the mean spend per customer is $ 135 - with a standard deviation of 55 - what balance should the company specify?

However, at the end of the first month it was found that 8% of customers qualified for the free gift. What has happened? Assuming that the standard deviation hasn’t changed, calculate the new mean spend per customer.

HOW do I calculate the Z Score for these questions ?

The first part of the question's z score = 1.645 which makes sense if looking at the z-table.

But according to the answer for the second part of the question the z score = 1.405. This I do not understand -- the closest percent on the z-table is 0.9207 with a z-score of 1.401 -- How do they calculate this ? and why is the closest percentile to 92% on the z-table not used ?

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You may be misreading the table. If it looks like the one below (from here), the closest value to $0.92$ is $0.9207$, which is at the intersection of the row for $z=1.4$ and the column $0.01$. This would correspond to a $z$-score of $1.4+0.01=1.41$, not $1.401$.

However, $0.92$ is nearly exactly between the values $0.9192$ ($z=1.40$) and $0.9207$ ($z=1.41$) in the table, so the $z$-score for $0.92$ would be roughly the average of $1.40$ and $1.41$, or $1.405$. The exact $z$-score above which $8\%$ of the normal distribution lies is (rounded to $6$ decimal places) $1.404071$ (see this calculator, for example)

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