According to the February 2008 Federal Trade Commission report on consumer fraud

Question

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.

Explanation / Answer

Below are the null and alternate hypothesis
H0: p = 0.23
Ha: p < 0.23

pcap = 321/1432 = 0.2242
SE = sqrt(0.23*0.77/1432) = 0.0111

Test statistic,
z = (0.2242 - 0.23)/0.0111
z = -0.5225

p-value = 0.3007

For the alpha = 0.05, we fail to reject the null hypothesis.
There are not significant evidence to conclude that Alaska had a lower proportion of identity theft.

Type I error would have occurred if we have incorrectly rejected the null hypothesis i.e. though the actual proportion is 0.23 but we concluded that it is less than 0.23.

Type II error would have occurred if we incorrectly fail to reject the null hypothesis i.e. though the actual proportion is less than 0.23 but we concluded to fail to reject the null hypothesis.

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