Main Answer:
The 99.7% confidence interval for the population mean, based on the sample, is 53.9% to 68.1%, indicating high confidence. A) 53.9%, 68.1%
Therefore, the correct answer is A) 53.9%, 68.1%.
Step-by-step explanation:
The 99.7% confidence interval for the population mean is between 53.9% and 68.1%. This interval indicates the range within which we are highly confident the true population mean lies based on the data from the random sample of 75 individuals, where 52 people prefer coffee to tea.
In statistical terms, a 99.7% confidence interval suggests that if we were to take many random samples and calculate the confidence interval for each, approximately 99.7% of those intervals would contain the true population mean. The calculation involves considering the variability in the sample data and the desired level of confidence.
The preference for coffee over tea in the given sample is reflected in the confidence interval's lower bound of 53.9%, indicating that at least 53.9% of the population prefers coffee. The upper bound of 68.1% suggests that, with high confidence, the proportion of people preferring coffee could be as high as 68.1%.
These confidence intervals are valuable in understanding the range of values that are likely to include the true population parameter, providing a more nuanced perspective beyond a single point estimate.
Therefore, the correct answer is A) 53.9%, 68.1%.