Question

Happy the elf (who is a male elf) tells you that one of the reasons he’s so happy is that the sex ratio of elves working at the North Pole is “two girls for every boy”2 . Slartibartfast, another elf, claims that Happy’s name should be “Delusional” because the proportion of female elves is much lower than Happy claims. You decide to test Happy’s claim by simply tallying the sex of each of the next 100 elves who walk by you on their way to work. How many male elves out of the 100 would it take for you to reject (at α = .05) Happy’s claim and find in favour of Slartibartfast? Show how you have calculated your answer.

Answer 1

Solution

“two girls for every boy” => proportion of boys, p = 2/3 = 0.6667.

The statistical equivalent of the given problem is: Find the critical value to reject the null hypothesis p = 2/3 against the alternative p < 2/3 at 5% significance level.

Let X = Number of boys in a sample of 100. Then, X ~ B(100, p)

Hypotheses:

Null H₀ : p = p₀ = 0.6667 [Happy’s claim] Vs H_A : p < 0.6667[claim of Slartibartfast]

Let t be the value of X below which H₀ is rejected. Then, by definition of significance level,

P(X < t under n = 100 and p = 0.6667) = 0.05 and our job is to find the value of t.

Now to find t,

The above probability calculation is to be done using Binomial probability. But, noting that both np = 66.67 and np(1 - p) = 22.22 are greater than 10, Binomial probability can be approximated by Standard Normal probability as explained below:

If X ~ B(n, p), np ≥ 10 and np(1 - p) ≥ 10, then Binomial probability can be approximated by Standard Normal probabilities by Z = (X – np)/√{np(1 - p)} ~ N(0, 1) i.e., P(X < > t) = P[Z < > {(t - np)/√{np(1 - p)}] where Z ~ N(0, 1)]

So,

P(X < t under n = 100 and p = 0.6667) = 0.05 is equivalent to

P[Z < {(t – 66.67)/√22.22} = 0.05

i.e., P[Z < {(t – 66.67)/4.7140} = 0.05

=> {(t – 66.67)/4.7140} = - 1.645 [using Excel function: Statistical NORMSINV]

=> t = 58.8187

Thus, the number of boys out of 100 is 58 or less is sufficient to reject (at α = .05) Happy’s claim and find in favour of Slartibartfast ANSWER

DONE

[going beyond,

Exact probability using Binomial for P(X < 58) = 0.0433 and P(X < 58) = 0.0658]