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A Simple Random Sample, often abbreviated as SRS, is a fundamental concept in statistics. It refers to a sampling method where each individual in the population has an equal chance of being selected.
This randomness ensures that the sample is unbiased and accurately represents the larger population.

• **Equal Chance**: Every member of the population should have the same probability of selection.
• **Unbiased Selection**: The selection should be free from systematic bias.

In the context of Latoya's survey, an SRS of 50 seniors was taken from a total population of 175. This process helps ensure that the sample accurately reflects the opinions of all seniors living in the dorm. SRS is crucial when estimating proportions because it supports the validity of statistical inferences, like confidence intervals, by minimizing potential biases.

The sample proportion, represented by the symbol \( \hat{p} \), is a critical measurement when estimating a population proportion. It indicates the fraction of the sample that has a particular characteristic.
To calculate it, you divide the number of successes (or occurrences of the desired outcome) by the total sample size.

• **Formula**: \( \hat{p} = \frac{\text{number of successes}}{\text{sample size}} \)
• **Example**: In Latoya's scenario, 14 out of 50 seniors like the cafeteria food, making \( \hat{p} = \frac{14}{50} = 0.28 \).

The sample proportion is a point estimate of the true population proportion and is used as a foundation for constructing confidence intervals. It helps to understand the likelihood of a certain proportion of the population sharing a specific attribute.

The Normality Condition ensures that the sample distribution of the proportion is approximately normal. This condition allows the use of normal probability models for accurate estimation. For the condition to hold:

Both \( n\hat{p} \) and \( n(1-\hat{p}) \)* should be at least 10.

This condition ensures the sample size is large enough for the sampling distribution to be normal.
In Latoya's case:

• **Calculate**: \( n = 50 \), \( \hat{p} = 0.28 \)
• **Values**: \( n\hat{p} = 50 \times 0.28 = 14 \); \( n(1-\hat{p}) = 50 \times 0.72 = 36 \)

Both numbers are above 10, fulfilling the Normality Condition.
This allows confidence intervals to be constructed using the normal distribution.

The Independence Condition is necessary to ensure that the sample observations do not influence one another. This helps maintain the validity of statistical inferences.
One way to check this condition is to ensure that the sample size is less than 10% of the entire population, which minimizes the chance of dependency.

• **Calculate**: The population size here is 175 seniors.
• **Condition**: For independence, the sample of 50 should be less than 175 \( \times 0.1 = 17.5 \).

Unfortunately, in Latoya's case, 50 exceeds 17.5, which suggests that the independence condition is not met.
This means that we need to be cautious about potential biases or dependencies that might arise during sampling. While this may add some uncertainty to the results, it doesn't render the findings invalid but highlights the importance of understanding sampling limitations.