Question: To test a manufacturer’s claim that 4 out of 5 dentists recommend brand K of toothpaste, a consumer-protection group randomly samples 400 dentists and asks them what they recommend. Of the sample, eighty-nine dentists did not recommend brand K while three-hundred-eleven did. Please answer the following questions.

a. What is the sample proportion of dentists that recommend brand K?

b. Based on the answer to (a) why would there even be a need to use statistical analysis to test the manufacturer’s claim in the first place? Why is it necessary?

c. Find a 90% confidence interval to estimate the true population proportion of dentists that recommend brand K.

d. What do you conclude about the manufacturer’s claim based on the confidence interval you found in (c)?

e. Let a null hypothesis be that the true population proportion of dentists that prefer brand K is 80% and an alternative hypothesis be that said proportion is less than 80%. Testing at the 5% significance level what should the consumer-protection group conclude?

f. Considering part (e) above, what is the probability that the statistician makes an error of Type 1?