Tangerine gives its banking customers a commission if they refer new people to open bank accounts at Tangerine. As a result, 55% of new customers at Tangerine came from referrals in 2014. Another Canadian bank had 14% referrals last year (out of a total of 65,000 new customers) and has increased the commission this year. They expect to achieve 20% referrals this year and pick a random selection of 400 new customers and find that 18% of them came from referrals.

a) Can we claim that the proportion of referrals has increased from last year at the 1% significance level?

b) Calculate a confidence interval in order to confirm the result in part (a). Does your confidence interval confirm that result?

c) Calculate the power of the test, based on the expected increase.

For (d)–(h), you are not asked to do the calculation, just to answer the question with reasons.

d) Supposing the result of our survey of 400 customers was that 21% came from referrals, would the answer to part (a) be different?

e) Supposing the result of our survey of 400 customers was that 21% came from referrals, would the power of our test be higher?

f) Supposing we used a 5% significance level, would the answer to part (a) be different?

g) Supposing we used a 5% significance level, would the power of our test be higher?

h) Supposing we sampled more than 400 customers, would the power be higher?