How far sharks swim
A sample of 151 female sharks has mean weight 103.6 kg. The 95% confidence interval for the mean weight was (102.1,\,105.1), so the margin of error is 1.5 kg.
A two-sample comparison of female and male mean weights has t=3.178 on 296 degrees of freedom. The fit of weekly swimming distance on weight and sex is
\widehat{\mathrm{distance}}=2.631+0.0937\times\mathrm{weight}-1.966\times\mathbf{1}_{\{\mathrm{male}\}},
with residual standard error 0.99 on 297 degrees of freedom.
How many females would be needed to cut the margin of error on the mean weight by a factor of four?
State the hypotheses for the two-sample test and the conclusion at level 0.05. The two-sided p-value is 0.00164.
Predict the distance for a 100 kg female and a 100 kg male.
The margin scales as 1/\sqrt n. Cutting it by four multiplies the sample size by 16, so 151\times 16=2416 females.
H_0 is equal mean weights. The two-sided alternative is that the means differ. The p-value 0.00164 rejects equality at level 0.05.
A 100 kg female is predicted to swim 2.631+0.0937\times 100=12.0 km. A 100 kg male is predicted to swim 12.0-1.966=10.0 km.