To determine how many teenagers should be included in the sample, we need to follow several steps to understand the distribution of the visitors across different ticket types and apply this to the sample size. Here's the detailed step-by-step solution:
1. Understanding Cumulative Frequencies and Actual Frequencies:
- Cumulative frequencies are given in the problem for different age categories:
- Less than 4 years old: 0 visitors
- Less than 12 years old: 70 visitors
- Less than 19 years old: 180 visitors
- Less than 59 years old: 300 visitors
- Using these cumulative frequencies, we can find the actual number of visitors in each age category relevant to the ticket types.
2. Calculating the Frequencies for Each Ticket Type:
- Child (Age 5-12): Since the cumulative frequency for less than 12 years is 70 and for less than 4 years is 0, the number of children is [tex]\( 70 - 0 = 70 \)[/tex].
- Teenager (Age 13-19): Since the cumulative frequency for less than 19 years is 180 and for less than 12 years is 70, the number of teenagers is [tex]\( 180 - 70 = 110 \)[/tex].
- Adult (Age 20-59): Since the cumulative frequency for less than 59 years is 300 and for less than 19 years is 180, the number of adults is [tex]\( 300 - 180 = 120 \)[/tex].
- Senior (Age 60+): The total number of visitors is 300 and the cumulative frequency for less than 59 years is 300, so the number of seniors is [tex]\( 300 - 300 = 0 \)[/tex].
3. Calculating the Proportion of Visitors Who Are Teenagers:
- The total number of visitors is 300.
- The number of teenagers is 110.
- The proportion of teenagers is given by [tex]\( \frac{\text{number of teenagers}}{\text{total number of visitors}} = \frac{110}{300} \approx 0.3667 \)[/tex].
4. Determining the Number of Teenagers in the Sample:
- The sample size is 30.
- To find the number of teenagers in the sample, we multiply the sample size by the proportion of teenagers:
[tex]\[
\text{number of teenagers in sample} = 30 \times 0.3667 \approx 11
\][/tex]
Therefore, the number of teenagers to be included in the sample is 11.
