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When dealing with statistical inference, it is often important to determine whether a sample proportion distribution can be approximated by a normal distribution. This is known as the normal approximation. To decide if this is possible, we apply the criteria:

• The condition \( np \geq 10 \)
• The condition \( n(1-p) \geq 10 \)

These criteria ensure the sample size is large enough so that the sampling distribution of the proportion is approximately normal. In the exercise provided, these conditions were used to assess whether the sample sizes of 60 and 50 might lead to a normal approximation.
However, the calculation \( np = 60 \times 0.08 = 4.8 \) reveals that the condition \( np \geq 10 \) is not met, indicating that the distribution cannot be approximated as normal. Understanding these conditions help in making inferences about population proportions using sample data.

The concept of sample proportion is fundamental in statistics when examining samples to make conclusions about a population. The sample proportion represents the fraction of the sample that has a certain characteristic. In our context, if 12% of a sample of students are vegetarians, the sample proportion \( \hat{p} \) is 0.12.
The sample proportion is used in various calculations, including determining the standard error and the z-score.

• It gives an estimate of the population proportion \( p \).
• It is crucial in hypothesis testing to check if observed proportions differ from assumed population proportions.

By consistently applying the concept of sample proportion, statisticians can estimate and infer the characteristics of an entire population based on sampled data.

Standard error is a measure of the dispersion or spread of the sample proportion, showing how much variation there is from the actual population proportion. It's calculated using the formula: \[SE = \sqrt{\frac{p(1-p)}{n}}\]where \( p \) is the population proportion, and \( n \) is the sample size.
The standard error decreases as the sample size increases, which represents a reduction in sampling variability.

• A smaller standard error indicates a more precise estimate of the population proportion.
• For larger sample sizes, the standard error is smaller, as observed in the calculation where the standard error for 125 students was reduced when the sample was increased to 250 students.

Understanding the standard error is key to assessing how sample statistics approximate the population parameters.

The z-score is a statistical measure that describes a value's position relative to the mean of a group of values. It helps determine how unusual or typical a sample statistic is under a given hypothesis. When comparing sample proportions to the population proportion, the z-score helps in deducing unusualness. It is calculated as:\[z = \frac{\hat{p} - p}{SE}\]where \( \hat{p} \) is the sample proportion, \( p \) is the population proportion, and \( SE \) is the standard error.
A z-score tells us how many standard deviations the sample proportion is from the population proportion.

• A z-score within the range of roughly \(-2\) to \(2\) suggests that the statistic is not unusual.
• A higher absolute value of the z-score indicates that the sample proportion is further from what would be expected if the null hypothesis were true.

In the exercise, z-scores were used to interpret if the observed proportion of vegetarians in larger samples was unusual, aiding decisions about statistical significance.