What is the level of significance? State the null and alternate hypotheses.

For the same data, null hypothesis, and level of significance, is it possible that a one-tailed test results in the conclusion to reject Hg while a two tilled test results in the conclusion to fail to reject Ho? Explain.

Interpret your conclusion in the context of the application.

Using the Standard Normal Table from the online lectures this week, what is the area under the standard normal curve:
a) To the left of a z-score of 1.25
b) To the right of a z-score of 1.25
c) Between the z-scores -0.25 and 0.55

A researcher suspects that the mean birth weights of babies whose mothers did not see a doctor before delivery is less than 3000 grams. The researcher states the hypotheses as H:x = 3000 grams H, x < 3000 grams.
These hypotheses are incorrect. What are the appropriate hypotheses for performing this significance test?
$H_0:\overline{x}>3000\text{ grams; }{H}_a:\overline{x}<3000\text{ grams, where }\overline{x}$ = the mean birth weights of the sample of babies whose mothers did not see a doctor before delivery.
$H_0:\overline{x}<3000\text{ grams; }{H}_a:\overline{x}=3000\text{ grams, where }\overline{x}$ = the mean birth weights of the sample of babies whose mothers did not see a doctor before delivery.
$H_0:\mu =3000\text{ grams; }{H}_a:\mu \ne 3000\text{ grams, where }\mu$ = the true mean birth weights of all babies whose mothers did not see a doctor before delivery.
$H_0:\mu =3000\text{ grams; }{H}_a:\mu <3000\text{ grams, where }\mu$ = the true mean birth weights of all babies whose mothers did not see a doctor before delivery.
$H_0:\overline{x}=3000\text{ grams; }{H}_a:\overline{x}\ne 3000\text{ grams, where }\overline{x}$ = the mean birth weights of the sample of babies whose mothers did not see a doctor before delivery.

Is Edwin Jaynes correct about statistics?
I've recently been reading Edwin Jaynes's book, Probability Theory: The Logic of Science, and I was struck by Jaynes's hostile view of what he dubs "orthodox statistics." He repeatedly claims that much of statistics is a giant mess, and argues that many commonplace statistical techniques are ad hoc devices which lack a strong theoretical footing. He blames historical giants like Karl Pearson and Ronald Fisher for the field's sorry state, and champions Bayesian methods as a healthier alternative.
From my personal perspective, his arguments make a lot of sense. However, there are a number of factors that prevent me from taking his criticism of statistics at face value. Despite the book's being published in 2003, the majority of its contents were written in the mid 20th century, making it a little dated. The field of statistics is reasonably young, and I'm willing to bet it's changed significantly since he levied his critiques.
Furthermore, I'm skeptical of how he paints statistics as having these giant methodological rifts between "frequentists" and "Bayesians." From my experience with other fields, serious methodological disagreement between professionals is almost nonexistent, and where it does exist it is often exaggerated. I'm also always skeptical of anyone who asserts that an entire field is corrupt--scientists and mathematicians are pretty intelligent people, and it's hard to believe that they were as clueless during Jaynes's lifetime as he claims.
Questions:
1. Can anyone tell me if Jaynes's criticisms of statistics were valid in the mid 20th century, and furthermore whether they are applicable to statistics in the present day? For example, do serious statisticians still refuse to assign probabilities to different hypotheses, merely because those probabilities don't correspond to any actual "frequencies?"
2. Are "frequentists" and "Bayesians" actual factions with strong disagreements about statistics, or is the conflict exaggerated?

A new cream that advertises 52% of women over the age of 50 will experience skin improvement from using their product. A study was conducted in which a sample of 170 women over the age of 50 used the new cream for 6 months. Of those 170 women, 76 of them reported skin improvement (as judged by a dermatologist). Is this evidence that the cream will improve the skin of less than 52 % of women over the age of 50? Perform a hypothesis test using the 0.05 level of significance.
Set up a null and alternate hypothesis.

According to shopper data, 97% of households purchase toilet paper. An analyst believes this is too low. To investigate, the analyst selects a random sample of 500 households and finds that 98% of them purchase toilet paper. We found that we do not have convincing evidence at the α = 0.05 level to conclude that the true proportion of all households that purchase toilet paper differs from 0.97.
The power of this test to reject p = 0.97 if 0.98 is true is 0.21. If the analyst were to decrease the significance level of the test, how would the power be affected?
It would increase.
It would decrease.
It would remain the same.
It would double.