Final Answer:
The correct range for the population mean, with 99.7% certainty, is between 47% and 55%, thus the correct option is b.
Step-by-step explanation:
In statistics, when estimating population parameters from a sample, we often use confidence intervals to express the range within which a population parameter is likely to lie. For this problem, we're estimating the proportion of individuals who prefer coffee to tea in the entire population based on the sample data (option b).
The sample proportion who prefer coffee is 52 out of 75 individuals. To estimate the population mean with 99.7% confidence, we use the standard deviation formula for proportions: σ = √[p(1 - p) / n], where p is the sample proportion and n is the sample size. Plugging the values, σ = √[(52/75) * (1 - 52/75) / 75] ≈ 0.0614.
For a 99.7% confidence interval, we use the empirical rule for normally distributed data, which states that approximately 99.7% of data lies within three standard deviations from the mean. Therefore, the population mean is estimated to be within ±3 standard deviations from the sample mean.
The population mean is the sample proportion ± 3 * σ. Thus, the range is 52/75 ± 3 * 0.0614 ≈ 0.52 ± 0.1842. Converting this to percentages, the range is approximately 47% to 55%, where 0.52 is equivalent to 52% and 0.1842 is equivalent to 18.42%. Therefore, the correct answer is (b) 47%, 55%. This range indicates that with 99.7% certainty, the proportion of individuals in the population who prefer coffee lies between 47% and 55%.