Question

1a. [1 mark] The price per kilogram of tomatoes, in euros, sold in various markets in a city is found to be normally distributed with a mean of 3.22 and a standard deviation of 0.84. Find the price that is two standard deviations above the mean price.

Answer 1

The price that is two standard deviations above the mean price is 4.90.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

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.

Mean of 3.22 and a standard deviation of 0.84.

This means that
`\mu = 3.22, \sigma = 0.84`

Find the price that is two standard deviations above the mean price.

This is X when Z = 2. So

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

`2 = (X - 3.22)/(0.84)`

`X - 3.22 = 2*0.84`

`X = 4.9`

The price that is two standard deviations above the mean price is 4.90.