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

We can model the number of dance club members in a sample of 36 students using a binomial distribution, where ( n = 36 ) and ( p = 0.08 ). Thus, we need to calculate:

$P(X = 4) = \binom{36}{4} (0.08)^4 (0.92)^{32}$

Calculating this gives approximately ( P(X = 4) \approx 0.167 ).

To find this probability, we start with:

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

We can calculate ( P(X \leq 7) ) using the binomial cumulative distribution function. After calculating, we find that:

$P(X > 7) \approx 0.0222$.

The probability that a student is a member of the university dance club and can dance the tango is the product of the individual probabilities. Given that 40% of dance club members can dance the tango:

$P(\text{dance club member and can dance tango}) = P(\text{dance club member}) \times P(\text{can dance tango | dance club member})$

This gives:

$0.08 \times 0.40 = 0.032.$

Using a binomial model with ( n = 50 ) and ( p = 0.032 ) for the number of students who are members of the dance club and can dance the tango:

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

Calculating each of these probabilities yields:

$P(X < 3) \approx 0.08.$