A political party is holding an election to choose a new leader. A statistician within the party decides to sample 70 party members to find their opinions of the leadership candidates. There are 4735 members under 30 years old and 8565 members 30 years old and over. The statistician wants to use a sample of 70 party members in the survey. He decides to use a random stratified sample. 15 (a) Calculate how many of each age group should be included in his sample. 15 (b) Explain how he could collect the random sample of members under 30 years old.

A-Level AQA Maths Pure Arithmetic Sequences & Series: A political party is holding an

A political party is holding an election to choose a new leader - AQA - A-Level Maths Pure - Question 15 - 2020 - Paper 3

Question 15

A political party is holding an election to choose a new leader. A statistician within the party decides to sample 70 party members to find their opinions of the le... show full transcript

Worked Solution & Example Answer:A political party is holding an election to choose a new leader - AQA - A-Level Maths Pure - Question 15 - 2020 - Paper 3

Step 1

Calculate how many of each age group should be included in his sample.

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Answer

To determine how many members from each age group should be sampled, we will use the formula for stratified sampling:

The total number of members is: $N = 4735 + 8565 = 13300$
The proportion of members under 30 years old is: $P_{<30} = \frac{4735}{13300}$
The expected sample size for members under 30 years old is: $n_{<30} = P_{<30} \times 70 = \frac{4735}{13300} \times 70 \approx 24.9 \approx 25$
The proportion of members 30 years old and over is: $P_{30+} = \frac{8565}{13300}$
The expected sample size for members 30 years old and over is: $n_{30+} = P_{30+} \times 70 = \frac{8565}{13300} \times 70 \approx 45.1 \approx 45$

Thus, approximately 25 members under 30 years old and 45 members 30 years old and over should be included in the sample.

Step 2

Explain how he could collect the random sample of members under 30 years old.

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Answer

To collect a random sample of members under 30 years old, the statistician can follow these steps:

Enumerated Population: Each member under 30 should be assigned a unique number from 1 to 4735, representing their position in the list of all eligible members.
Random Selection: Use a random number generator or calculator to produce random numbers within the range of 1 to 4735.
Selection of Members: The statistician should continue to generate random numbers until 25 unique numbers have been identified. Each number corresponds to a member under 30 years old.
Replacement: If a number corresponds to a member who has already been selected, the statistician should generate a new number, ensuring every member is chosen only once.
Documentation: The members represented by the selected numbers should be documented for easy identification and follow-ups.

A political party is holding an election to choose a new leader - AQA - A-Level Maths Pure - Question 15 - 2020 - Paper 3

Worked Solution & Example Answer:A political party is holding an election to choose a new leader - AQA - A-Level Maths Pure - Question 15 - 2020 - Paper 3

Calculate how many of each age group should be included in his sample.

Explain how he could collect the random sample of members under 30 years old.

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