Question

The sales data for January and February of a frozen yogurt shop are approximately normal. The mean daily sales for January was $300 with a standard deviation of $20. On the 15th of January, the shop sold $310 of yogurt. The mean daily sales for February was $320 with a standard deviation of $50. On the 15th of February, the shop sold $340 of yogurt. Which month had a higher z-score for sales on the 15th, and what is the value of that z-score

Answer 1

January had a higher z-score for sales on the 15th, and the value of that z-score was of 0.5.

Explanation:

z-score:

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

January:

The mean daily sales for January was $300 with a standard deviation of $20. On the 15th of January, the shop sold $310 of yogurt. This means, respectively, that
`\mu = 300, \sigma = 20, X = 310`. So

`Z = (X - \mu)/(\sigma)`

`Z = (310 - 300)/(20)`

`Z = 0.5`

February:

The mean daily sales for February was $320 with a standard deviation of $50. On the 15th of February, the shop sold $340 of yogurt. This means, respectively, that
`\mu = 320, \sigma = 50, X = 340`. So

`Z = (X - \mu)/(\sigma)`

`Z = (340 - 320)/(50)`

`Z = 0.4`

January had a higher z-score for sales on the 15th, and the value of that z-score was of 0.5.