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Tonya's goal is to determine how many seniors from her school intend to attend the prom.
In statistical terms, she is trying to establish information about a certain population.
Here, the **population of interest** consists of all the senior students at Tonya's school. In total, there are 750 such students. Her research aims to find out about the whole group using a statistical technique.
The **parameter of interest**, on the other hand, is the specific aspect Tonya wants to measure or infer about this population. In this case, she wants to estimate the proportion of seniors planning to go to the prom.
This proportion is denoted by the symbol \( p \). Essentially, understanding \( p \) would tell us what portion of the entire senior class is fun and party-ready.

One crucial requirement for making predictions about the whole population is that the sample must be random.
Tonya achieved this by selecting a simple random sample (SRS) of 50 students from the 750 seniors.
By ensuring that every senior had an equal chance of being chosen, her sample accurately reflects the diversity and behavior of the entire population.
A random sample helps in reducing biases and represents different sub-groups within the population.
With a fair sample selection process, there's increased confidence that the insights derived from this sample hold true for the entire group of seniors.

• Ensures fair and unbiased selection
• Enhances the accuracy of the study
• Reflects the overall population's variation

Thus, Tonya's strategy of random sampling plays a pivotal role in making her findings reliable.

When estimating proportions, it's vital to ensure that the sample size and expected frequencies uphold the **Normality Condition**.
This condition states that both \( np \) and \( n(1-p) \) should be greater than 10. This ensures the sample data is sufficiently large enough to approximate a normal distribution, which is essential for constructing confidence intervals.In Tonya's case, the sample size \( n = 50 \) and the sample proportion \( \hat{p} = 0.72 \).
Therefore, \( np = 50 \times 0.72 = 36 \) and \( n(1-p) = 50 \times 0.28 = 14 \). Both values are indeed greater than 10.
This implies that her sample satisfies the Normality Condition, providing a solid foundation for Tonya to estimate the proportion of seniors heading to the prom using a confidence interval.

• Ensures the sample size is adequate
• Allows approximation to a normal distribution
• Builds a foundation for reliable interval estimation

The **10% Condition** is another crucial factor to check when creating a confidence interval. This ensures that the sample size should be less than 10% of the population size, keeping the sampling process independent.In Tonya's situation, she sampled 50 seniors out of the total 750.
This gives us a percentage of \( \frac{50}{750} \times 100 \% = 6.67\% \), which is clearly under the 10% threshold.
By meeting the 10% Condition, Tonya can maintain the assumption of independence between each selected student in her sample.

• Keeps the sample size within a reasonable range
• Ensures independence among sample measurements
• Ensures the reliability of the statistical inferences made

This vital check assures that her subsequent analyses, like the construction of a confidence interval, are legitimate and sound.