Question

Laser scanning for fish volume estimation. engineers design tanks for rearing commercial fish to minimize both the use of natural resources (water) and the rearing volume necessary to ensure fish welfare.
One key to a well-designed tank is obtaining a reliable estimate of the volume (biomass) of fish reared in the tank.
The feasibility of a laser scanning technique for estimating fish biomass was investigated in the Journal of Aquacultural Engineering (Nov.2012). Fifty turbot fish were reared in a tank for experimental purposes.
A laser scan was executed in four randomly selected locations in the tank and the volume (in kilograms) of fish layer at each location was measured.
The four laser scans yielded a mean volume of 240 kg with a standard deviation of 15 kg. from this information, estimate the true mean volume of fish layer in the tank with 99% confidence. Interpret the result, practically.

What assumption about the data is necessary for the inference derived from the analysis to be valid?

Answer 1

Answer:

The 99% confidence interval would be given by (196.2;283.8)

We are 99% condident that the true mean is between 196.2 and 283.8  

We need to assume that the data comes from a random sample and we need to assume that the distribution of the data is normal.  

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X=240$ represent the sample mean for the sample  

$\mu$ population mean (variable of interest)

s=15 represent the sample standard deviation

n=4 represent the sample size  

Part a

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=4-1=3$

Since the Confidence is 0.99 or 99%, the value of $\alpha=0.01$ and $\alpha/2 =0.005$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,3)".And we see that $t_{\alpha/2}=5.84$

Now we have everything in order to replace into formula (1):

$240-5.84\frac{15}{\sqrt{4}}=196.2$    

$240+5.84\frac{15}{\sqrt{4}}=283.8$

So on this case the 99% confidence interval would be given by (196.2;283.8)    

We are 99% condident that the true mean is between 196.2 and 283.8

What assumption about the data is necessary for the inference derived from the analysis to be valid?

We need to assume that the data comes from a random sample and we need to assume that the distribution of the data is normal.