5. The proportion of all blood recipients stricken with viral hepatitis is known

Question

5. The proportion of all blood recipients stricken with viral hepatitis is known to be 0.07. Suppose a new treatment is believed to reduce the incidence rate. This treatment is given to 200 blood recipients a. If the true proportion is 0.07, describe the sampling distribution of making sure to comment on its center, spread, and shape and referencing any theorems you may have used. b. Under the new treatment only 6 of the 200 patients contract hepatitis. This appears to be a good result, and the question of interest to medical researchers is: Does this result indicate that the true (long- run) proportion of patients who contract hepatitis after the experimental treatment is less than 0.07 (indicating the new treatment is effective) or could this result be due to sampling variability? Provide an answer to the researchers making sure to justify your answer statistically.

Explanation / Answer

Part (a)

Let p0 be the true population proportion and pcap be the proportion obtained based on a random sample of size n. Then, test statistic to comment of the population proportion is

Z = (pcap - p0)/{ p0(1 - p0)/n}.

The sampling distribution of Z is asymptotically (i.e., as n tends to infinity or when n is large enough) Normal with mean 0 and variance 1 (Standard Normal Distribution). ANSWER

Part (b)

Test

Let X = Number of patients in the sample contracting hepatitis.

Then, X ~ B(n, p), where n = sample size and p = probability of a patient contracting hepatitis which is also equal to the proportion of patients contracting hepatitis in the population.

Claim :

The new treatment is effective

Hypotheses:

Null H0 : p = p0 = 0.07 Vs HA : p < 0.07

Test Statistic:

Z = (pcap - p0)/{p0(1 - p0)/n} where pcap = sample proportion and n = sample size.

Calculations:

Z = {(6/200) – 0.07}/{ 0.07(0.93)/200}

= - 2.217

Distribution, Critical Value and p-value:

Under H0, distribution of Z can be approximated by Standard Normal Distribution, provided

np0 and np0(1 - p0) are both greater than 10.

So, given a level of significance of %, Critical Value = lower % of N(0, 1), and

p-value = P(Z < Zcal).

Since the question has not specified a level of significance, we will take it as 5%, the most popular and oft-employed level.

So, Critical Value = - 1.645 [using Excel Function]    

p-value = P(Z < - 2.217) = 0.0133

Decision Criterion (Rejection Region):

Reject H0, if Zcal < Zcrit or if p-value < .

Decision:

Since Zcal < Zcrit, H0 is rejected which is also confirmed by the p-value < .

=> H1 is accepted => the proportion is less than 0.07.

Conclusion :

There is enough evidence to suggest that the claim that the new treatment is effective is valid.

DONE

Related Questions

Navigate

• Browse (All)
• Browse 5
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.