Question

In each problem show all steps of the hypothesis test. If some of the assumptions are not met, note that the results of the test may not be correct and then continue the process of the hypothesis test.

1.Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a given time period and found that 11% of all lenses had defects of some type. Looking at the type of defects, they found in a three-month time period that out of 34,641 defective lenses, 5865 were due to scratches. Are there more defects from scratches than from all other causes? Use a 1% level of significance.

2. According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Arkansas had 1,601 complaints of identity theft out of 3,482 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Arkansas had a higher proportion of identity theft than 23%? Test at the 5% level.

3. In 2001, the Gallup poll found that 81% of American adults believed that there was a conspiracy in the death of President Kennedy. In 2013, the Gallup poll asked 1,039 American adults if they believe there was a conspiracy in the assassination, and found that 634 believe there was a conspiracy ("Gallup news service," 2013). Do the data show that the proportion of Americans who believe in this conspiracy has decreased? Test at the 1% level.

Answer 1

1) The null and alternative hypothesis

Note : This is right tail test as we want to if proportion is more than 11%

Test statistic is

where

5865/34641= 0.169

Thus ,

= 35.10

At 1% level of significance , one tailed (right tailed ) critical value of z is

zc = 2.33

Rejection region is in the right tail , reject H0 if z > 2.33

Since calculated z > 2.33

We reject H0

At 1% level of significance there is sufficient evidence to conclude that there are more defects from scratches than other causes.

2) The null and alternative hypothesis

Note : This is right tail test as we want to if proportion is more than 23%

Test statistic is

where

1601/3482= 0.46

Thus ,

= 32.25

At 5% level of significance , one tailed critical value of z is

zc =1 .65

Rejection region is in the right tail , reject H0 if z > 1.65

Since calculated z > 1.65

We reject H0

At 5% level of significance there is sufficient evidence to conclude that Arkansas had a higher proportion of identity theft than 23% .

3) The null and alternative hypothesis

Note : This is left tail test as we want to if proportion is less than 81%

Test statistic is

where

634/1039 =0.61

Thus ,

= -1.64

At 1% level of significance , one tailed (left tailed)  critical value of z is

zc = - 2.33

Rejection region is in the left tail , that is reject H0 if z < - 2.33

Since calculated value of z > - 2.33

We fail to reject H0

At 1% level of significance there is not sufficient evidence to conclude that proportion of Americans who believe in this conspiracy has decreased .