Question

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 133 of the 270 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 0.05 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

(a) No, based on the above information, it cannot be said that the % of 12th graders who believe that prescription opioids are easily available as this is only a sample of students from the population. Another sample could show that the % has not changes, whereas yet another sample could show an increase.

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(b) The Hypothesis:

H0: p = 0.54

Ha: p < 0.54

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(c) In this case we are using a left tailed test. Depending on what we have to hypothesis, we either use a left or a right tailed test which are 1 tailed tests. We look out for the key words, less than, reduced or decreased for a left tailed test and for a right tail test we see if we have been asked to check if a quantity has increased (more than and larger being other key words). The alternative hypothesis of a left tailed test has a '<' sign and the right tailed will have a ' >' sign.

The other option is a 2 tailed test, which we used when we have to see if a value is 'different' from the one that has been hypothesized. Sometimes the question simply states, we want to see if a certain quantity is equal to 'x'. This means that we have to do a 2 tailed test. A 2 tailed test will always contain the not equal to ' ' sign in the alternative hypothesis.

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(d) A Type I error is the rejection of a true null hypothesis. In this case it means we reject the claim that 54% of the 12th grade who believe that prescription opioids are easily available when actually it is 54% and conclude that the % is less than 54%.

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(e) A Type II error is the failure to reject a false null hypothesis. In this case it would mean that we reject the claim that the % of the 12th grade who believe that prescription opioids are easily available is less than 54% and conclude that the % is equal to 54%.

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(f) At = 0.05, for a left tail test the critical value = -1.645

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(g) The Test Statistic: = 133/270 = 0.4926

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(h) The p Value: The p value (Left tail) for Z = -1.56, is; p value = 0.0594

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(i) 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 (-1.56) is > -Zcritical (-1.645), We Fail to Reject the Null Hypothesis.

The p value Method: Since P value (0.0594) is > (0.05), We Fail to Reject the Null Hypothesis.

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(j) The Conclusion:   There is insufficient evidence at the 95% significance level to conclude that the % of 12th graders who believe that prescription opioids are easily available has reduced since 2010.

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