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The sample proportion is a statistical measure used to estimate the proportion of a specific characteristic in a population based on a sample.To calculate the sample proportion, simply divide the number of successes (people who recognized and respected the athlete) by the total number of observations (total customers surveyed).In our case, the sample proportion (denoted as \( \hat{P} \)) is calculated as follows:

• Number of successes: 16
• Sample size: 100
• Sample proportion: \( \hat{P} = \frac{16}{100} = 0.16 \)

This means that 16% of the sample recognizes and respects the woman golfer.The sample proportion is an essential part of many statistical analyses, as it gives a direct insight into the sample data relative to the entire population being studied.

Standard error is a key concept in statistics that gives us an idea of the variability in the sample proportion.It is calculated to understand how much the sample proportion (\( \hat{P} \)) can be expected to differ from the true population proportion.In simple terms, the standard error helps gauge the reliability of the sample proportion used to estimate the population parameter.To calculate the standard error of the sample proportion, use the formula: \[ SE = \sqrt{\frac{\hat{P} \times (1 - \hat{P})}{n}} \]For the golfer’s scenario:

• \( \hat{P} = 0.16 \)
• Sample size \( n = 100 \)
• \( SE = \sqrt{\frac{0.16 \times 0.84}{100}} \approx 0.037 \)

This calculation shows that the variability in the sample is approximately 0.037, indicating how much the sample proportion could vary if different samples were taken from the population.

The Z-score is a statistical measure that shows how many standard deviations an element is from the mean.In our context, it helps us determine the range of values that capture the true population proportion with a certain level of confidence.The Z-score is crucial when calculating confidence intervals, especially in determining how far the sample proportion could be from the true proportion.For a 95% confidence interval, a typical Z-score value used is 1.96.This information is typically obtained from a Z-table, which is derived from the standard normal distribution.Using the formula for the confidence interval: \( \hat{P} \pm Z \times SE \), where:

• \( \hat{P} \) is the sample proportion (0.16),
• \( Z = 1.96 \),
• \( SE = 0.037 \).

The confidence interval provides a range in which the true population proportion is likely to lie, using the Z-score to account for variability.

Statistical analysis involves using statistical methods to collect, examine, and interpret data to make informed decisions. In the context of the advertising campaign exercise, statistical analysis helps determine the unknown true population proportion of customers who recognize and respect the golfer based on sample data. Here, we utilize the following steps in the analysis:

• Calculate the sample proportion
• Determine the standard error to understand the variability of the sample proportion
• Use the Z-score for a chosen confidence level (here, 95%)
• Construct a confidence interval for the true proportion

Conducting this statistical analysis allows the company to make data-driven decisions. They can confidently predict the success of their advertising campaign based on the sample data and respective computations.