One of the truly impressive applications of statistics occurred in connection with the design of the 1954 Salk polio-vaccine experiment and analysis of the resulting data. Part of the experiment focused on the efficacy of the vaccine in combating paralytic polio. Because it was thought that without a control group of children, there would be no sound basis for assessment of the vaccine, it was decided to administer the vaccine to one group and a placebo injection (visually indistinguishable from the vaccine but known to have no effect) to a control group. For ethical reasons and also because it was thought that the knowledge of vaccine administration might have an effect on treatment and diagnosis, the experiment was conducted in a double-blind manner. That is, neither the individuals receiving injections nor those administering them actually knew who was receiving vaccine and who was receiving the placebo (samples were numerically coded). (Remember: at that point it was not at all clear whether the vaccine was beneficial.) Let p1 and p2 be the probabilities of a child getting paralytic polio for the control and treatment conditions, respectively. The objective was to test H0: p1 − p2 0 versus Ha: p1 − p2 0 (the alternative states that a vaccinated child is less likely to contract polio than an unvaccinated child). Supposing the true value of p1 is 0.0003 (an incidence rate of 30 per 100,000), the vaccine would be a significant improvement if the incidence rate was halved—that is, p2 = . Using a level ? = 0.05 test, it would then be reasonable to ask for sample sizes for which ? = 0.1 when p1 = 0.0003 and p2 = 0.00015. Assuming equal sample sizes, the required n is obtained as n = 1.645 (0.5)(0.00045)(1.99955) + 1.28 (0.00015)(0.99985) + (0.0003)(0.9997) 2 (0.0003 − 0.00015)2 = (0.0349 + 0.0271)/ 2 ≈ 171,000 The actual data for this experiment follows. Sample sizes of approximately 200,000 were used. The reader can easily verify that z = 6.43—a highly significant value. The vaccine was judged a resounding success! Placebo: m = 201,229, x = number of cases of paralytic polio = 110 Vaccine: n = 200,745, y = 33

One of the truly impressive applications of statistics occurred in connection with the design of the 1954 Salk polio-vaccine experiment and analysis of the resulting data. Part of the experiment focused on the efficacy of the vaccine in combating paralytic polio. Because it was thought that without a control group of children, there would be no sound basis for assessment of the vaccine, it was decided to administer the vaccine to one group and a placebo injection (visually indistinguishable from the vaccine but known to have no effect) to a control group. For ethical reasons and also because it was thought that the knowledge of vaccine administration might have an effect on treatment and diagnosis, the experiment was conducted in a double-blind manner. That is, neither the individuals receiving injections nor those administering them actually knew who was receiving vaccine and who was receiving the placebo (samples were numerically coded). (Remember: at that point it was not at all clear whether the vaccine was beneficial.) Let p1 and p2 be the probabilities of a child getting paralytic polio for the control and treatment conditions, respectively. The objective was to test H0: p1 − p2 0 versus Ha: p1 − p2 0 (the alternative states that a vaccinated child is less likely to contract polio than an unvaccinated child). Supposing the true value of p1 is 0.0003 (an incidence rate of 30 per 100,000), the vaccine would be a significant improvement if the incidence rate was halved—that is, p2 = . Using a level ? = 0.05 test, it would then be reasonable to ask for sample sizes for which ? = 0.1 when p1 = 0.0003 and p2 = 0.00015. Assuming equal sample sizes, the required n is obtained as n = 1.645 (0.5)(0.00045)(1.99955) + 1.28 (0.00015)(0.99985) + (0.0003)(0.9997) 2 (0.0003 − 0.00015)2 = (0.0349 + 0.0271)/ 2 ≈ 171,000 The actual data for this experiment follows. Sample sizes of approximately 200,000 were used. The reader can easily verify that z = 6.43—a highly significant value. The vaccine was judged a resounding success! Placebo: m = 201,229, x = number of cases of paralytic polio = 110 Vaccine: n = 200,745, y = 33

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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One of the truly impressive applications of statistics occurred in connection with the design of the 1954 Salk polio-vaccine experiment and analysis of the resulting data. Part of the experiment focused on the efficacy of the vaccine in combating paralytic polio. Because it was thought that without a control group of children, there would be no sound basis for assessment of the vaccine, it was decided to administer the vaccine to one group and a placebo injection (visually indistinguishable from the vaccine but known to have no effect) to a control group. For ethical reasons and also because it was thought that the knowledge of vaccine administration might have an effect on treatment and diagnosis, the experiment was conducted in a double-blind manner. That is, neither the individuals receiving injections nor those administering them actually knew who was receiving vaccine and who was receiving the placebo (samples were numerically coded). (Remember: at that point it was not at all clear whether the vaccine was beneficial.)
Let p₁ and p₂ be the probabilities of a child getting paralytic polio for the control and treatment conditions, respectively. The objective was to test H₀: p₁ − p₂ 0 versus H_a: p₁ − p₂ 0 (the alternative states that a vaccinated child is less likely to contract polio than an unvaccinated child). Supposing the true value of p₁ is 0.0003 (an incidence rate of 30 per 100,000), the vaccine would be a significant improvement if the incidence rate was halved—that is, p₂ = . Using a level ? = 0.05 test, it would then be reasonable to ask for sample sizes for which ? = 0.1 when p₁ = 0.0003 and p₂ = 0.00015. Assuming equal sample sizes, the required n is obtained as

1.645

(0.5)(0.00045)(1.99955)

+ 1.28

(0.00015)(0.99985) + (0.0003)(0.9997)

	2

(0.0003 − 0.00015)²

(0.0349 + 0.0271)/

	2

≈ 171,000

The actual data for this experiment follows. Sample sizes of approximately 200,000 were used. The reader can easily verify that z = 6.43—a highly significant value. The vaccine was judged a resounding success!

Placebo: m	= 201,229, x = number of cases of paralytic polio = 110
Vaccine: n	= 200,745, y = 33

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