1. Part 1: What are the conditions for using the standard deviation formula when conducting a significance test? Be specific about p versus p-hat. 

Part 2: What are the conditions for approximating with a normal distribution?

2. In each of the following cases state whether or not the normal approximation to the binomial should be used for a significance test on the population proportion p. 

a. n = 10 and H0: p = 0.4. 

b. n = 100 and H0: p = 0.6. 

c. n = 1000 and H0: p = 0.996. 

d. n = 500 and H0: p = 0.3.

3. Leroy, a starting player for a major college basketball team, made only 38.4% of his free throws last season. During the summer he worked on developing a softer shot in the hope of improving his free-throw accuracy. In the first eight games of this season Leroy made 25 free throws in 40 attempts. Let p be his probability of making each free throw he shoots this season.

a. State the null hypothesis H0 that Leroy's free-throw probability has remained the same as last year and the alternative Ha that his work in the summer resulted in a higher probability of success.

b. Calculate the z statistic for testing H0 versus Ha.

c. Do you accept or reject H0 for a = 0.05? Find the P-value.

d. Give a 90% confidence interval for Leroy's free-throw success probability for the new season. Are you convinced that he is now a better free-throw shooter than last season?

e. What assumptions are needed for the validity of the test and confidence interval calculations that you performed?

4. A software company is planning for an upgrade of their software. You must charge customers $100. Are your customers willing to pay this much? You contact a random sample of 40 customers and find that 11 would pay $100 for the upgrade. If the upgrade is to be profitable, you will need to sell it to more than 20% of your customers. Do the sample data give good evidence that more than 20% are willing to buy?

a. Formulate this problem as a hypothesis test. Give the null and alternative hypotheses.

b. Carry out the significance test. Report the test statistic and the P-value.

c. Should you proceed with plans to produce and market the upgrade?