Question

State the Type I and Type II errors in complete sentences given the following statements.

a. The mean number of years Americans work before retiring is 34.

b. At most 60% of Americans vote in presidential elections.

c. The mean starting salary for San Jose State University graduates is at least \(100,000 per year.

d. Twenty-nine percent of high school seniors get drunk each month.

e. Fewer than 5% of adults ride the bus to work in Los Angeles.

f. The mean number of cars a person owns in his or her lifetime is not more than ten.

g. About half of Americans prefer to live away from cities, given the choice.

h. Europeans have a mean paid vacation each year of six weeks.

i. The chance of developing breast cancer is under 11% for women.

j. Private universities mean tuition cost is more than \)20,000 per year.

Short answer

a. Type I error: We conclude that the mean is not 34 years when it really is 34 years. Type II error: We conclude that the mean is 34 years, when in fact it really is not 34 years.

b. Type I error: We conclude that more than 60% of Americans vote in presidential elections when the actual percentage is at most 60%. Type II error: We conclude that at most 60% of Americans vote in presidential elections when, in fact, more than 60% do.

c. Type I error: We conclude that the mean starting salary is less than $100,000 when it really is at least $100,000. Type II error: We conclude that the mean starting salary is at least $100,000 when, in fact, it is less than $100,000.

d. Type I error: We conclude that the proportion of high school seniors who get drunk each month is not 29% when it really is 29%. Type II error: We conclude that the proportion of high school seniors who get drunk each month is 29% when, in fact, it is not 29%.

e. Type I error: We conclude that fewer than 5% of adults ride the bus to work in Los Angeles when the percentage that does is really 5% or more. Type II error: We conclude that 5% or more adults ride the bus to work in Los Angeles when, in fact, fewer than5% do.

f. Type I error: We conclude that the mean number of cars a person owns in his or her lifetime is more than 10 when in reality it is not more than 10. Type II error: We conclude that the mean number of cars a person owns in his or her lifetime is not more than 10 when, in fact, it is more than 10.

g. Type I error: We conclude that the proportion of Americans who prefer to live away from cities is not about half, though the actual proportion is about half. Type II error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half.

h. Type I error: We conclude that the duration of paid vacations each year for Europeans is not six weeks, when in fact it is six weeks. Type II error: We conclude that the duration of paid vacations each year for Europeans is six weeks when, in fact, it is not.

i. Type I error: We conclude that the proportion is less than 11% when it is really at least 11%. Type II error: We conclude that the proportion of women who develop breast cancer is at least 11%, when in fact it is less than 11%.

j. Type I error: We conclude that the average tuition cost at private universities is more than $20,000, though, in reality, it is at most $20,000. Type II error: We conclude that the average tuition cost at private universities is at most $20,000 when, in fact, it is more than $20,000.

Step-by-step solution

Part (a) Solution: 

Type I error: We conclude that the mean is not 34 years when it really is 34 years. Type II error: We conclude that the mean is 34 years, when in fact it really is not 34 years. A two-tailed test would be required here.

Part (b) Solution: .

Type I error: We conclude that more than 60% of Americans vote in presidential elections when the actual percentage is at most 60%. Type II error: We conclude that at most 60% of Americans vote in presidential elections when, in fact, more than 60% do Type 1 and Type 2 errors should be carefully taken care of.

Part (c) Solution: 

Type I error: We conclude that the mean starting salary is less than $100,000 when it really is at least $100,000. Type II error: We conclude that the mean starting salary is at least $100,000 when, in fact, it is less than $100,000. Type 1 and Type 2 errors should be carefully taken care of.

Part (d) Solution:

Type I error: We conclude that the proportion of high school seniors who get drunk each month is not 29% when it really is 29%. Type II error: We conclude that the proportion of high school seniors who get drunk each month is 29% when, in fact, it is not 29%. Type 1 and Type 2 errors should be carefully taken care of.

Part (e) Solution: 

Type I error: We conclude that fewer than 5% of adults ride the bus to work in Los Angeles when the percentage that does is really 5% or more. Type II error: We conclude that 5% or more adults ride the bus to work in Los Angeles when, in fact, fewer than5% do. Type 1 and Type 2 errors should be carefully taken care of.

Part (f) Solution: 

Type I error: We conclude that the mean number of cars a person owns in his or her lifetime is more than 10 when in reality it is not more than 10. Type II error: We conclude that the mean number of cars a person owns in his or her lifetime is not more than 10 when, in fact, it is more than 10. Type 1 and Type 2 errors should be carefully taken care of.

Part (g) Solution: 

Type I error: We conclude that the proportion of Americans who prefer to live away from cities is not about half, though the actual proportion is about half. Type II error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half. Type 1 and Type 2 errors should be carefully taken care of.

Part (h) Solution:

Type I error: We conclude that the duration of paid vacations each year for Europeans is not six weeks, when in fact it is six weeks. Type II error: We conclude that the duration of paid vacations each year for Europeans is six weeks when, in fact, it is not. Type 1 and Type 2 errors should be carefully taken care of.

Part (i) Solution: 

Type I error: We conclude that the proportion is less than 11% when it is really at least 11%. Type II error: We conclude that the proportion of women who develop breast cancer is at least 11%, when in fact it is less than 11%. Type 1 and Type 2 errors should be carefully taken care of.

Part (j) Solution: 

Type I error: We conclude that the average tuition cost at private universities is more than $20,000, though, in reality, it is at most $20,000. Type II error: We conclude that the average tuition cost at private universities is at most $20,000 when, in fact, it is more than $20,000. Type 1 and Type 2 errors should be carefully taken care of.