A marine biologist is monitoring the daily migration speeds of two species of shark, Species P P\,P and Species QQQ. The biologist believes that the mean speed of Species Q Q\,Q is more than 1.0 km/day1.0\text{ km/day}1.0 km/day faster than the mean speed of Species PPP.
A random sample of 400 sharks of Species P P\,P and 50 sharks of Species Q Q\,Q was tracked. The results are summarized in the table below.
Shark SpeciesSample size (n)Sample mean speed (vˉ)Unbiased variance estimate (s2)Species P40032.425Species Q5034.616 \begin{array}{|c|c|c|c|} \hline \text{Shark Species} & \text{Sample size } (n) & \text{Sample mean speed } (\bar{v}) & \text{Unbiased variance estimate } (s^2) \\ \hline \text{Species } P & 400 & 32.4 & 25 \\ \hline \text{Species } Q & 50 & 34.6 & 16 \\ \hline \end{array} Shark SpeciesSpecies PSpecies QSample size (n)40050Sample mean speed (vˉ)32.434.6Unbiased variance estimate (s2)2516Test, at the 5% level of significance, the biologist's claim. State your hypotheses, test statistic, critical value, and conclusion clearly.
State two assumptions required for the validity of the test in part (a).
The sample for Species Q Q\,Q is later increased to 51 sharks. The daily migration speed of the additional shark is recorded as 55 km/day55\text{ km/day}55 km/day.
Find an unbiased estimate of the variance for the expanded sample of 51 sharks of Species QQQ.