91Ó°ÊÓ

When working with statistics, the sample proportion is a crucial element. In this example, it helps us understand what fraction of our sample possesses a certain characteristic. Here, the sample consists of 205 college students, and we want to find out how many of them believe in haunted places. Out of these 205 students, 65 believe in haunted places.

To find the sample proportion, we use the formula:

• \( \hat{p} = \frac{\text{Number of favorable outcomes}}{\text{Sample size}} \)

For our scenario, it is:

• \( \hat{p} = \frac{65}{205} \approx 0.3171 \)

This value indicates that approximately 31.71% of the sampled students believe in haunted places. The sample proportion is a simple yet powerful tool for estimations and forming hypotheses about a population.

The standard error of the sample proportion plays a vital role in determining how much our sample proportion may vary from the true population proportion. It's a measure of the variability or "spread" of the proportion estimate.

To calculate the standard error for a proportion, we use the formula:

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

Here, the following values are used:

• \( \hat{p} = 0.3171 \)
• \( n = 205 \)

Substituting these values, we get:

• \[ SE = \sqrt{\frac{0.3171 \times (1-0.3171)}{205}} \approx 0.0336 \]

This estimated standard error tells us about the variability of the sample proportion. A smaller standard error would mean that our sample proportion is a more precise estimate of the true population proportion.

The z-score is a statistical metric that helps us understand how far a given data point is from the mean in terms of standard deviations. This is crucial when constructing confidence intervals, especially when we want to predict the true proportion of a population based on a sample.

In our case, we want a 99% confidence interval, which aligns with the z-score value. The z-score represents the number of standard deviations a result is from the mean.

• For a 99% confidence level, the z-score is approximately 2.576.

This value suggests that we're looking for an interval that covers 99% of the possible sample proportions, centered around the sample proportion we computed. The z-score helps us determine how wide this interval should be.

Statistical tables are extremely handy tools when calculating confidence intervals and other statistical metrics. They contain pre-calculated values for probabilities and statistical tests, which are used to find critical values, like the z-score, without complex computations.

In our scenario, we used the statistical table to find the z-score corresponding to a 99% confidence interval. Without these tables, determining such values would require intricate calculations.

The key takeaways when using statistical tables include:

• Understand which table to use: Z-table for normal distributions, t-table for t-distributions, etc.
• Identify the correct row and column based on your requirement
• Ensure use of the accurate critical value for your desired level of confidence

These tables simplify and streamline the process, allowing us to readily find critical statistical values, making them invaluable in statistical adjustments and inferences.