Common scenario for all questions: The investigators plan to use a sample of size 60 to carry out tests of hypotheses with the null hypothesis =250=250.

They think the population standard deviation is equal to 50.

They really want their hypothesis tests to correctly reject the null hypothesis if =235=235. Use R power.t.test from the pwr package or similar functions.

Assume that the population mean really is 265. What size sample would be needed to have 90% power to correctly reject the null hypothesis =250=250 if the investigators use a one-sided alternative hypothesis >250>250, using =0.05=0.05? Continue to use 50 as an estimate of the population standard deviation. Show how you found the answer.

Assume that the population mean really is 265. What size sample would be needed to have 90% power to correctly reject the null hypothesis =250=250 if the investigators use a two-sided alternative hypothesis 250=250, using =0.05=0.05? Continue to use 50 as an estimate of the population standard deviation. Show how you found the answer.

Compare your answers to questions 5.4 and 5.5 and comment on the difference. Include specifics and relevant results.