Question

In November 2012, a research study claimed that less than two-thirds of U.S. adults thought that President Obama would keep U.S. safe from terrorists during his second term. In a random sample of 1,009 adults, 655 stated that they believed this.

Test the study's claim at 5% significance that less than two-thirds (67%) of U.S. adults thought that Obama would keep the U.S. safe from terrorists during his second term.

A. Step 1: Determine the Hypotheses

B. Step 2: Collect the Data

C. Step 3: Assess the Evidence

D. Step 4: State a Conclusion

5. Think about the possibility that your conclusion in the hypothesis is in error.

A. Suppose we drew the wrong conclusion in the hypothesis test above on Obama and terrorists. What type of error would this be?

B. Explain what must be true about the population for such an error to occur.

6. The test statistic in the last problem did not lead us to support the claim that less than two-thirds of U.S. adults believed that Obama would protect U.S. from terrorists in his second term.

A. Suppose that the sample size was doubled, and from this sample, the same sample proportion was observed. Would the new statistic make us more or likely to support the claim?

B. Does increasing the sample size make the difference between the sample proportion and the assumed population proportion more statistically significant, real-world significant, or both?

Answer 1

a)

Ho :   p =    0.67                  
H1 :   p <   0.67       (Left tail test)          
                          
Level of Significance,   α =    0.05                  
Number of Items of Interest,   x =   655                  
Sample Size,   n =    1009                  
                          
Sample Proportion ,    p̂ = x/n =    0.6492                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0148                  
Z Test Statistic = ( p̂-p)/SE = (   0.6492   -   0.67   ) /   0.0148   =   -1.4080
                          
  
p-Value   =   0.0796 [excel function =NORMSDIST(z)]              
Decision:   p value>α ,do not reject null hypothesis

conclusion; there is not sufficient evidence to say that   less than two-thirds (67%) of U.S. adults thought that Obama would keep the U.S. safe from terrorists during his second term.   

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5)

A)

we are accepting the null hypothesis while it is false

so it would be type II error

6)

A)

If we double the sample size, Standard error will decrease, so test stat will be more significant leading the test result significant

So , yes the new statistic will make us more or likely to support the claim

B)

increasing the sample size make the difference between the sample proportion and the assumed population proportion more statistically significant