Question

7.1.6

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, Alaska had 321 complaints of identity theft out of 1,432 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? State the type I and type II errors in this case, consequences of each error type for this situation, and the appropriate alpha level to use.

Answer 1

Since is not mentioned, we use = 0.05

p = 0.23, = 321 / 1432 = 0.2242

The Hypothesis:

H0: p 0.23: The proportion of identity thefts in Alaska is greater than or equal to 0.23.

Ha: p < 0.23: The proportion of identity thefts in Alaska is lesser than 0.23.

This is a Left tailed Test.

The Test Statistic:

The p Value:    The p value (Left tail) for Z = -0.52, is; p value = 0.3015

The Critical Value:   The critical value (Left tail) at = 0.05, Zcritical = -1.645

The Decision Rule:

The Critical Value Method: If Zobserved is <- Zcritical Then Reject H0.

The p value Method: If the P value is < , Then Reject H0

The Decision:   

The Critical Value Method: Since Z observed (-0.52) is > -Zcritical (-1.645), We Fail to Reject H0.

The p value Method: Since P value (0.3015) is > (0.05), We Fail to Reject H0.

The Conclusion:   There is insufficient evidence at the 95% significance level to conclude that the proportion of identity thefts in Alaska is lesser than 0.23.

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Type I Error: The incorrect rejection of a true null hypothesis. It means that the null hypothesis, p = 0.23 is true, but the same is rejected and we accept the alternative hypothesis , p < 0.23.

The Consequence here is that identity authorities might feel that the identity thefts have reduced and not introduce new methods to tackle the same.

Type II Error: The incorrect acceptance of a false null hypothesis. It means that the null hypothesis, p = 0.23 is false, but the same is accepted and we reject the alternative hypothesis , p < 0.23.

The Consequence here is that identity authorities might feel that the identity thefts have increased or that their efforts are being wasted and invest more of the taxpayers money than what is neccesary to curb the threat.