Question

Gribbles are small, pale white, marine worms that bore through wood. While sometimes considered a pest since they can wreck wooden docks and piers, they are now being studied to determine whether the enzyme they secrete will allow us to turn inedible wood and plant waste into biofuel. \({ }^{25}\) A sample of 50 gribbles finds an average length of \(3.1 \mathrm{~mm}\) with a standard deviation of \(0.72 .\) Give a best estimate for the length of gribbles, a margin of error for this estimate (with \(95 \%\) confidence), and a \(95 \%\) confidence interval. Interpret the confidence interval in context. What do we have to assume about the sample in order to have confidence in our estimate?

Short answer

From the calculations, best estimate for the length of gribbles is 3.1 mm. The margin of error for this estimate at 95% confidence level would be calculated based on SE and the Z-value for 95% confidence level. The 95% confidence interval would be the range calculated by subtracting and adding the margin of error from the mean. This interval gives us an estimation of where the true population mean lies with 95% confidence. The assumptions are that the sample is a simple random sample representative of the population and the population distribution is approximately normal.

Step-by-step solution

Calculate Standard Error (SE)

The standard error can be calculated using the formula: SE = \(\frac{s}{\sqrt{n}}\) where \(s\) is the standard deviation and \(n\) is the size of the sample. In this case \(s = 0.72\) and \(n = 50\). Plug these values into the formula to find the SE.

Calculate Margin of Error (ME)

The margin of error can be calculated using the formula: ME = SE * Z where the Z value corresponds to the desired confidence level. For a 95% confidence interval, the Z value is 1.96. Multiply the SE calculated in step 1 by 1.96 to find the ME.

Calculate 95% Confidence Interval

A confidence interval can be calculated using the formula: CI = mean ± ME. The mean given in the problem is 3.1. Add and subtract the ME calculated in step 2 from the mean to find the confidence interval.

Interpret the Confidence Interval

The confidence interval calculated in step 3 is an estimate of the range in which the true population parameter (the length of a gribble) lies with a 95% level of confidence. It means, if the experiment were repeated many times, 95% of the intervals would contain the true population parameter.

State the Assumptions

To have confidence in our estimate, we must assume that the sample of gribbles is a simple random sample - it represents a portion of the entire population of gribbles without any bias. Also, we assume that the distribution of gribble length in the population is approximately normal.