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

The random variable X is modeled by a binomial distribution B(36, 0.08) since there are 36 students and the probability of each student being in the dance club is 0.08. We can calculate P(X = 4) using the formula:

P(X = k) = {n race k} p^k (1-p)^{n-k}

Here,

• n = 36 (total trials)
• k = 4 (successes)
• p = 0.08 (success probability)

Calculating:

P(X = 4) = {36 race 4} (0.08)^4 (1 - 0.08)^{36 - 4}
This evaluates to approximately 0.167.

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

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

Calculating P(X ≤ 7) involves summing individual probabilities from P(X = 0) to P(X = 7):

$P(X > 7) = 1 - \\sum_{k=0}^7 P(X = k)$
This can be computed, yielding an approximate value of 0.222.

The probability a student is a member of the dance club and can dance the tango is given by the product of the probability of being in the dance club and the probability of being able to dance the tango. With 40% of dance club members being able to dance the tango, we have:

$P(Membership \\text{ and } Tango) = P(Membership) imes P(Tango | Membership)$

Thus, $P = 0.08 imes 0.40 = 0.032.$

For this part, we can use a binomial distribution B(50, 0.032) since we take a sample of 50 students. The required probability is:

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

Calculating these probabilities and summing gives the total probability for fewer than 3 members. It is approximately 0.08.