Question

5.            In 2016, over 42,000 people were killed by opioid overdoses. The effects of the issue are not limited to fatalities. An additional issue is the lack of proper development among young adolescent users during a critical phase of brain maturation. One method of reducing opioid abuse is to reduce the availability of prescription opioids.   In 2010, 54 percent of students in 12th grade believed that prescription opioids were easily accessible. In a 2017 study, one agency surveyed three high schools in the northeast and found 266 of the 500 12th grade students surveyed believed prescription opioids were easily accessible in their community.

                a.            Is the above information sufficient for you to be certain that the percentage of all 12th graders who believe prescription opioids are easily accessible has declined? Why or why not?

                b.            In establishing a statistical hypothesis testing of this situation, give the required null and alternative hypotheses for a test to determine if the percent of 12th graders who believe opioids are easily accessible has declined from 2010.

                                                H0:

                                                H1:

                c.             Based on your answer in part b, should you use a right-tailed, a left-tailed, or a two-tailed test? Briefly explain how one determines which of the three possibilities is to be used.            

                d.            Describe the possible Type I error for this situation--make sure to state the error in terms of the percent of 12th graders and their beliefs about opioid accessibility.                                          

                e.            Describe the possible Type II error for this situation--make sure to state the error in terms of the percent of 12th graders and their beliefs about opioid accessibility.          

                f.             Determine the appropriate critical value(s) for this situation given a 0.01 significance level.           

                g.            Determine/calculate the value of the sample's test statistic.         

                h.            Determine the P-value.                                                                

                i.              Based upon your work above, should you "Reject the null hypothesis" or "Fail to reject the null hypothesis?" Explain why.

                j.             Based upon your work above (and overlooking the flaws in the survey method), is there statistically sufficient evidence in this sample to support the claim that the percent of 12th graders who believe opioids are easily accessible has declined from 2010? Briefly explain your reasoning.

Answer 1

n =   500
x =    266
p̄ = x/n = 0.532

a) No the above information is not sufficient to be certain that the percentage of all 12th graders who believe prescription opioids are easily accessible has declined. As this is the result of only one sample. another sample may show different results.

b) Null and Alternative hypothesis:  
Ho : p =   0.54
H1 : p <   0.54

c) We should use a left-tailed test.

when we have to check the difference then two tailed test is used.

for less or declined we use left tailed test

For more or increased or higher we use Right tailed test.

d) Type I error is rejecting the null hypothesis when it is true.

In this context, type I error is rejecting the claim that the 54 percent of 12th graders who believe opioids are easily accessible, when in reality it is 54 percent.

e) Type II error is failing to reject the null hypothesis when it is false

In this context, type II error is failing to reject the claim that the 54 percent of 12th graders who believe opioids are easily accessible, when in reality it is less than 54 percent.

f) Critical value :

At α =    0.01   , left tailed critical value, z crit = NORM.S.INV(0.01   ) =   -2.326

g) Test statistic:  
z =(p̄ -p)/(√(p*(1-p)/n))  = (0.532 -0.54)/(√(0.54*(1-0.54)/500)) = -0.3589

h) p-value:

p-value = NORM.S.DIST( -0.3589 , 1) = 0.3598

i) As p-value = 0.3598> 0.01, we fail to reject the null hypothesis.

j) No, there is not statistically sufficient evidence in this sample to support the claim that the percent of 12th graders who believe opioids are easily accessible has declined from 2010.