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

Given that the number of members in the dance club is 8%, we can model X using a binomial distribution:

$X \sim B(36, 0.08)$

To find P(X = 4), we use the binomial probability formula:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Here, n = 36, k = 4, and p = 0.08:

$P(X = 4) = \binom{36}{4} (0.08)^4 (0.92)^{32} \approx 0.167$

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

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

We will calculate P(X ≤ 7) using the cumulative distribution function (CDF) for the binomial distribution. This involves summing individual probabilities from P(X = 0) to P(X = 7):

After calculating:

$P(X > 7) \approx 0.222$

We know that 40% of the dance club members can dance the tango. Therefore, the probability of a student being a member of the dance club and can dance the tango is:

$P(dance \ club \ and \ tango) = P(dance \ club) \times P(dance \ tango | dance \ club) = 0.08 \times 0.4 = 0.032$

Consider that in a sample of 50 students, we are interested in those who are members of the dance club and can dance the tango, with a probability of 0.032.

Using a binomial distribution:

Let Y be the number of students:

$Y \sim B(50, 0.032)$

We want to find P(Y < 3), which can be calculated as:

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

After calculation, we find:

$P(Y < 3) \approx 0.08$