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1. The Census Bureau is planning to take a sample amounting to 1/10 of 1% of the population in each state in order to estimate the percentage of the population in that state earning over $50,000 a year. Other things being equal, which of the following is true (circle one):

(i) The accuracy to be expected in California (population thirty million) is about the same as the accuracy to be expected in Nevada (population one million).

(ii) The accuracy to be expected in California is quite a bit higher than in Nevada.

(iii) The accuracy to be expected in California is quite a bit lower than in Nevada.

Please explain your reasoning.

THE CORRECT CHOICE IS (ii)
The accuracy in California will be quite a bit higher due to the the much larger sample size in that state (30,000) as compared with Nevada (1,000). Statistical accuracy doesn't rely on the percentage of the population (which is the same for each state) but it relies instead on the total number sampled.

Or look at the problem another way -- a sample of 1,000 for each state would yield the same accuracy even though as a percentage it is much smaller for California.

2. Extensive research conducted during the 1970s demonstrated that pregnancy durations are normally distributed with a mean of 266 days and a standard deviation of 16 days. In a very recent study of 16 pregnancy durations selected at random, the average pregnancy duration was 286 days with standard deviation of 20 days. Does this new study provide evidence that pregnancy durations have increased since the 1970s?

State the null hypothesis and the alternative hypothesis.

Null: average is equal to 266
Alternative: average is greater than 266

Perform a test of significance and state the resulting p-value. What would you conclude?

The appropriate test here is the Z because the standard deviation of the population is known. You might think of it as already existing at the time the new sample is drawn.

Test is

         286 - 266
      ------------------  = Z = 20/4 = 5
      (SQRT(16) x 16) / 16

It looks like pregnancy durations are getting longer. A Z of 5 suggests than the p-value or the probability is a lot less than 1%. In other words, if the true parameter is 266, the chance of drawing a sample of 16 and getting an average of 286 is very remote. This is evidence than the null is not correct.