Question

Continue the example in which the farmer claims that organic tomatoes are higher in vitamin A than conventional tomatoes, and the hypotheses to be tested are Ho : y = 1500 versus HQ:p > 1500, where u is the average amount of viiamin A (in IU) in organically grown tomatoes. Let's assume that the distribution of vitamin A is symmetric, so that the median amount of vitamin A in conventional tomatoes is also 1500 IU. In other words, half of the tomatoes have more than 1500 IU of vitamin A, and half have less. Suppose the farmer will select four organic tomatoes "at random" and reject Ho if all of the tomatoes have greater than 1500 IU of vitamin A. (a) What is the test size a for this decision rule? (b) Suppose the alternative is true and, in particular, suppose 80% of organic tomatoes have more than 1500 IU of vitamin A. What is the power of the test?

Answer 1

(a)

We have to calculate size of the test i.e. probability of type-I error.

Suppose, random variable X denotes the amount of vitamin A (in IU) in an organically grown tomato.

It is given that, the distribution of X is symmetric about its median 1500 IU, which is mean also (due to symmetric nature).

:: P(X < 1500) = P(X > 1500) = 0.5

The rejection region W is defined as 'reject H₀ if all of the four tomatoes have greater than 1500 IU of vitamin A'.

Clearly, amount of vitamin in a tomato is independent of other tomatoes.

:.a = P(X EW|H) = 0.54 = 0.0625

(b)

Our rejection region W is defined as 'reject H₀ if all of the four tomatoes have greater than 1500 IU of vitamin A'.

When alternative hypothesis H_a is true, probability of amount of vitamin A in a tomato more than 1500 is 0.8 i.e. the probability to fall in the rejection region.

:. power = P(X EW|H2) = 0.84 = 0.4096