Question

Tommy has a bag with 7 marbles in it, and decides to draw 3 marbles from the bag randomly. The marbles in the bag are all the same, but some of the marbles in the bag are red and some are purple. If we decide to test the null hypothesis "More red marbles than purple marbles" and the alternative hypothesis "More purple marbles than red marbles", what would be the level of significance? We will make the Rejection Region the event of getting atleast 2 marbles that are purple.

Answer 1

Here as hypothesises are given which are

null hypothesis "More red marbles than purple marbles"

alternative hypothesis "More purple marbles than red marbles"

so here we have to find the significance level that means the probability that we conclude that null hypothesis is wrong even if it is correct.

or in simple words, in relity there are more red marbles than purple marbles but we get atleast 2 marbles that are purple

so the combination of red (R) and purple (P) marbles where red marbles are more than purple marbles are (4R, 3P), (5R,2P), (6R, 1P) and (7R , 0P)

Here there are 8 combinations of Red and purple marbles so probability of each combination happening is 1/8

Now we have to find probability that atleast 2 marbles are purple

Pr(at least two marble are purple out of 3) = Pr(when 4 red and 3 purple marble are there) + Pr(when 5Red and 2 purple marble are there)

Pr(at least two marble are purple out of 3 when 4 red and 3 purple marble are there) = Pr(2 purple marble out of 3) + Pr(3 purple marble out of 3)

= ³C₂ * ⁴C₁/ ⁷C₃ + ³C₃ * ⁴C₀ /⁷C₃

= 12/35 + 1/35 = 13/35

Pr(two purple marble when 5Red and 2 purple marble are there) = ⁵C₀ * ²C₂/⁷C₃ = 1/35

so significance level = (13/35 + 1/35) * 1/8 = 14/35 * 1/8 = = 0.05