State one disadvantage of using quota sampling compared with simple random sampling - Edexcel - A-Level Maths Statistics - Question 1 - 2021 - Paper 1

Since X follows a binomial distribution where n = 36 and p = 0.08, we can calculate P(X = 4) as follows:

P(X = 4) = inom{36}{4} (0.08)^4 (0.92)^{32}

Calculating this gives:

= rac{36!}{4!(36-4)!} (0.08)^4 (0.92)^{32} = 0.167387

Thus, P(X = 4) is approximately 0.167.

To find P(X > 7), we can use the complement rule:

$P(X > 7) = 1 - P(X \\leq 7)$

We calculate P(X ≤ 7) using a binomial distribution and sum up from 0 to 7:

$P(X ≤ 7) = ext{sum of } P(X = k) ext{ for } k = 0 ext{ to } 7$

Using binomial calculations or a software gives us:

$P(X > 7) = 1 - 0.22223 = 0.77777$

Therefore, P(X > 7) is approximately 0.222.

Let T be the event that a student can dance the tango. Given that only 40% of the dance club members can dance the tango, we have:

$P( ext{dance club and can dance the tango}) = P( ext{dance club}) imes P(T | ext{dance club})$

Thus,

$P( ext{dance club}) = 0.08$ $P(T | ext{dance club}) = 0.40$

Therefore,

$P( ext{dance club and can dance the tango}) = 0.08 imes 0.40 = 0.032$

Using the binomial distribution with n = 50 and p = 0.032, we find:

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$

Calculating these, we have:

P(X = k) = inom{50}{k} (0.032)^k (0.968)^{50-k}

So,

$P(X = 0) + P(X = 1) + P(X = 2) ext{ gives us the desired probability.}$

Using the calculations, we find an approximate value of 0.08.