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. The type I error is not thinking mean as $34$and type II error is thinking mean as $34$

b. The type I error says at most $60\%$American's vote and type II error is more than $60\%$Americans vote.

c. The type I error is mean starting salary is less than $\$100,000$and type II error is at least $\$100,000$

d. The type I error says $29\%$seniors does not drinking and type II error is they do.

e. Type I error is fewer ride bus to work and type II error is more ride bus to work.

f. Type I error is that the mean number of cars a person owns in his or her lifetime is more than and type II error is the mean number of cars a person owns in his or her lifetime is not more than $10.$

g. Type I error is the proportion of Americans who prefer to live away from cities is not about half and type II error is the proportion of Americans who prefer to live away from cities is about half.

h. Type I error is the duration of paid vacations each year for Europeans is not$6$ weeks and type II error the duration of paid vacations each year for Europeans is $6$ weeks.

i. Type I error is the proportion is less than $11\%$ and type II error is the proportion is at least $11\%$.

j. Type I error is the proportion is less than $20,000$ and type II error is the proportion is at most $20,000$.

Step-by-step solution

Introduction

The one-sample t-test is a statistical hypothesis test that is used to evaluate if an unknown population mean differs from a given value.

Explanation (Part a)

Rejecting the null hypothesis $H_0$when it is true is defined as a $TypeI$error:

If the $H_0$is: "The mean number of years Americans work before retiring is $34$," than $TypeI$error is

• One thinks the mean is not $34$years when, in fact, it really is $34$years.
• $TypeII$error is One thinks the mean is $34$years when, in fact, it really is not $34$years.

Explanation (Part b)

Type I error: We conclude that more than $60\%$Americans vote in presidential elections when the actual percentage is at most $60\%$.

Type II error: We conclude that most $60\%$Americans vote in presidential elections when, in fact, more than $60\%$do.

Explanation (Part c)

$TypeI$error: One thinks the mean starting salary is less than $100,000$, in fact, it really is at least $100,000$

$TypeII$error: We conclude that the mean starting salary is at least $100,000$when, in fact, it really is less than $100,000$

Explanation (Part d)

$TypeI$error: We conclude that the proportion of high school seniors who get drunk each month is not $29\%$, when it really is $29\%$.

$TypeII$ error: We conclude that the proportion of high school seniors who get drunk each month is $29\%$when, in fact, it is not $29\%$

Explanation (Part e)

$TypeI$error: One thinks fewer than $5\%$adults ride the bus to work in Los Angeles when the percentage that does is real $5\%$or more.

$TypeII$error: One thinks more adults $5\%$ride bus tow work.

Explanation (Part f)

$TypeI$error: One thinks 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 $10$.

$TypeII$error: One thinks the mean number of cars a person owns in his or her lifetime is not more than $10$when, in fact, it really is more than $10$.

Explanation (Part g)

$TypeI$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.

$TypeII$error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half.

Step 9:  Explanation (Part h)

$TypeI$error: We conclude that the duration of paid vacations each year for Europeans is not six weeks when it really is six weeks.

$TypeII$error: We conclude that the duration of paid vacations each year for Europeans is six weeks when, in fact, it really is not.

Explanation (Part i)

$TypeI$ error: One thinks the proportion is less than $11\%$when it is really at least $11\%$.

$TypeII$ error: One thinks the proportion of women who develop breast cancer is at least $11\%$when it is really less than $11\%$.

Step 11:  Explanation (Part j)

$TypeI$error: One thinks the average tuition cost at private universities is more than $20,000$, though in reality, it is at most $20,000$

$TypeII$error: One thinks that the average tuition cost at private universities is at most $20,000$when, in fact, it is more than $20,000$

The final result state that, Type $I$and Type $II$errors in complete sentences given statements.