91Ó°ÊÓ

The sample proportion, often denoted as \( \hat{p} \), is a simple yet powerful statistical measure. It helps us understand how frequently a particular characteristic occurs within a sample. To compute the sample proportion, you divide the number of occurrences of the characteristic by the total sample size. For example, in a sample size of 2400 individuals, if 420 possess the characteristic of interest, the sample proportion is calculated as:\[ \hat{p} = \frac{420}{2400} = 0.175 \]This value, 0.175 in this case, indicates that 17.5% of the sampled population demonstrates the characteristic. Sample proportion helps in estimating the proportion in the larger population based on our sample data.

Population proportion, denoted as \( p \), is a value that gives us insight into how common a particular characteristic is within the entire population rather than just a sample. While it's often unknown, we try to estimate it using the sample proportion. To do this accurately, we construct confidence intervals which provide a range that likely contains the true population proportion. The calculations involve the sample proportion and also consider the sample size, because larger samples tend to provide better estimates of the population proportion.

The Z-score is a crucial part of constructing confidence intervals for population proportions. It tells us how many standard deviations a point is from the mean of a standard normal distribution. When constructing confidence intervals, the Z-score indicates the range of values we will use to ensure our interval has the desired level of confidence, such as 90% or 99%. For example:

• A 90% confidence interval uses a Z-score of 1.645.
• A 99% confidence interval uses a Z-score of 2.576.

By multiplying the Z-score with the standard error of the sample proportion, we expand the range of possible values for the true population proportion.

Determining whether a sample size is adequate is pivotal for constructing meaningful confidence intervals. We need to confirm that both \( n \hat{p} \) and \( n(1-\hat{p}) \) are at least 10. This ensures that the sample size is large enough for the normal approximation to be valid. In our example:

• \( n \hat{p} = 2400 \times 0.175 = 420 \)
• \( n(1-\hat{p}) = 2400 \times 0.825 = 1980 \)

Both values exceed 10, confirming our sample size is adequate. Adequate sample size is crucial for the reliability of any statistical inference or estimate drawn from sample data.