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The sample proportion is a key concept in statistics that captures the portion of a sample that possesses a specific characteristic. It serves as an estimate of the population proportion. To find the sample proportion, use the formula:

• \( \hat{p} = \frac{x}{n} \)

Here, \( x \) represents the number of successes (seeds that germinate), and \( n \) is the total number of trials (seeds planted).
In the example provided, 180 seeds out of 200 germinated. Plug these values into the formula:

• \( \hat{p} = \frac{180}{200} = 0.90 \)

This means that 90% of the seeds germinated based on your sample. You estimate the same for the population, knowing there is inherent sample variation.

The standard error of the sample proportion measures how much you expect the sample proportion to vary from the true population proportion. It assesses the variability of your estimate due to random sampling.
The standard error is calculated using the formula:

• \( SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} \)

This formula considers both the sample proportion and the sample size. In the seed example, you calculate the standard error as:

• \( SE = \sqrt{\frac{0.90 \times 0.10}{200}} \approx 0.0212 \)

A smaller standard error indicates that your sample proportion is a more precise estimate of the true population proportion. A larger sample size generally leads to a smaller standard error, improving the precision of your confidence interval.

The Z-score is crucial when determining confidence intervals because it accounts for the distribution you're working with. For the normal distribution, specific Z-scores correspond to particular confidence levels. At a 95% confidence level, the Z-score is usually 1.96 due to the properties of the normal distribution.
The Z-score tells you how many standard deviations your sample proportion is away from the mean under the standard normal distribution. To adjust the sample proportion into a confidence interval, you multiply the standard error by the Z-score and add it to, and subtract it from, the sample proportion.

• Example: \( \hat{p} \pm Z \times SE = 0.90 \pm 1.96 \times 0.0212 \)

This calculation provides a range of values, offering an estimate of where the true population proportion likely falls.

The normal distribution is a fundamental principle in statistics, often known for its bell-shaped curve. It is essential for understanding how data tends to be distributed around the mean.
When calculating confidence intervals, the normal distribution is used because it allows you to make inferences about the mean and variability of a population from a sample. The Z-score standardizes the sample proportion relative to this distribution, making confidence intervals possible.
In the context of the seed germination problem, assuming a normal distribution allows the use of Z-scores and simplifies complex statistical calculations, helping provide a reliable interval for the true proportion of seeds expected to germinate. Despite not all data being perfectly normal, large sample sizes often make the approximation valid, a principle established by the Central Limit Theorem.