Q#99: Estimating the confidence interval for the length of an angelfish
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Suppose you’re working on a study to understand the length of an average angelfish in the Amazon River. The study draws a random sample of 1,000 angelfish from an estimated population size of 2,500,000. Each fish in the sample of 1,000 is measured and recorded. We find that the average length in our sample is 2.5 inches and the standard deviation is 0.25 inches.
Given this information, calculate the 95% confidence interval for this sample and explain what the interval means.
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ANSWER
In the vast ecosystem of the Amazon River, angelfish play a crucial role, … yep backstory not as important, we are data scientists have to do Statistics only :).
Calculating the 95% Confidence Interval:
The formula for calculating the confidence interval for a population mean is:
Confidence Interval=Sample Mean±(Critical Value×Standard Error)Confidence Interval=Sample Mean±(Critical Value×Standard Error)
Where:
- Sample Mean = 2.5 inches
- Critical Value (for a 95% confidence interval) ≈ 1.96 (obtained from standard normal distribution tables)
- Standard Error = Standard Deviation/sqrt(Sample Size)
Given values:
- Standard Deviation = 0.25 inches
- Sample Size = 1,000
SE=0.25/sqrt(10000)≈0.0079
Now, substitute the values into the formula:
Confidence Interval=2.5±(1.96×0.0079)Confidence Interval=2.5±(1.96×0.0079)
Confidence Interval=2.5±0.0155
The 95% confidence interval is approximately (2.4845,2.5155)(2.4845,2.5155) inches.
Interpreting the Confidence Interval:
A 95% confidence interval means that if we were to conduct this study numerous times and construct a confidence interval each time, we would expect 95% of those intervals to contain the true average length of angelfish in the Amazon River.
In our specific study, we are 95% confident that the true average length of angelfish in the Amazon River lies between 2.4845 inches and 2.5155 inches. This interval provides a range of plausible values, acknowledging the variability inherent in taking a sample rather than measuring the entire population.
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