11C In-Class Activity

The Federal Trade Commission (FTC) is a bipartisan federal agency in the United  States. It was created in 1914 under President Woodrow Wilson. Its goal is to enforce  “federal consumer protection laws that prevent fraud, deception, and unfair business  practices.”[1]

Credit: iStock/lldo Frazao

Question 1

Do you think it is important to have these protections in place?

Question 2

In the preview assignment, you were told that in 2020, the national percentage of  complaints to the FTC due to identity theft was 29.4%. A commission in Florida is  asked to study the complaints locally and to determine whether Florida is exceeding  the national trend. Complete the following table to understand which observed  sample proportions might be unusual.

Number of  complaints  due to

identity

theft (out of  500)

 Value of [latex]\hat{p}[/latex], the sample

proportion

 [latex]z = \frac{\hat{p}-0.294}{0.0204}[/latex] P-value Do you think  we have

convincing

evidence to

suggest that  Florida is

exceeding the  national

trend? Why?
148 0.296 [latex]z[/latex] = 0.098 0.461 No, because a  sample

proportion of  0.296 is not

that unlikely

given the

national trend  of 0.294.
150
155
160
165
170

Question 3

At what point does it appear that something unusual or unexpected is happening in  Florida? That is, how many identity theft claims out of 500 total claims would make  you think that Florida is exceeding the national trend?

Suppose that in a sample of 500 claims, 460 of them were because of identity theft. If  the true population proportion is really 0.294, is it reasonable to observe a random  sample proportion of [latex]\hat{p} = \frac{460}{500} = 0.92[/latex]? Was this particular group of 500 people just  incredibly unusual OR could it be that the population proportion is something else,  which allows us to reject the null hypothesis?

A P-value assists us in determining whether or not we have evidence to reject the null  hypothesis. In statistics, we establish a “cut-off” value. There is no absolute cut-off  value, but typically we use 5%.

This 5% represents the extreme areas under the curve, which means they represent  unusual values. We compare the P-value to [latex]\alpha[/latex], which is the significance level of the  test. The significance level, [latex]\alpha[/latex], is the cut-off for P-values at which we have enough  evidence to reject the null hypothesis. Typically, small significance levels such as 1%,  5%, or 10% are used in hypothesis testing. You will learn more about significance levels  and their importance in In-Class Activity 11.E.

In order to make a claim about the null hypothesis, we write [latex]\alpha[/latex] as a decimal and  compare it to the P-value, as follows:

  • If P-value [latex]\leq \alpha[/latex], we have enough evidence to reject the null hypothesis, and  we have convincing evidence to support the alternative hypothesis.
  • Otherwise, we fail to reject the null hypothesis or do not reject the null  hypothesis, and we do NOT have convincing evidence to support the  alternative hypothesis.
    • When we fail to reject a null hypothesis, it does not mean there is  support in favor of the null hypothesis. Instead, this means that we just  did not see enough evidence to be convinced that the null hypothesis  is not true.

Question 4

Suppose that the Florida commission observes that out of a random sample of 500  claims filed with the FTC, 176 of them are due to identity theft. At the 5%  significance level, is there enough evidence to suggest that Florida is in fact  exceeding the national trend? Note that we already checked the conditions for a  one-sample z-test.

  1. Calculate the sample proportion.
  2. Write the null hypothesis.
  3. Write the alternative hypothesis.
  4. Calculate the test statistic.
  5. Using the DCMP Normal Distribution tool at https://dcmathpathways.shinyapps.io/NormalDist/, calculate the P-value. In  other words, identify the area on the right of your test statistic.
  6. At the 5% significance level, is there enough evidence to reject the null  hypothesis? Explain.
  7. At the 5% significance level, is there convincing enough evidence to suggest  that Florida is in fact exceeding the national trend? Explain.

 

  1. Federal Trade Commission. (n.d.). Enforcement. https://www.ftc.gov/enforcement ↵