Question

1. In a clinical study, 4200 subjects were vaccinated with a flu vaccine manufactured by growing cells in fertilized chicken eggs. Over a period of roughly 28 weeks, 32 of these subjects developed the flu.

You must do all of your calculations on the TI calculator. DO NOT Use the Stats -> Tests for any of the problems.

a. Find the observed statistic. Use the proper notation ( x or p ˆ ). [2pts]

b. Verify that the sample size is large enough to use the normal distribution to construct a confidence interval for the proportion of the population that were vaccinated with the vaccine but still developed the flu. [2 pts]

c. Construct a 95% confidence interval for the proportion of the population that were vaccinated with the vaccine but still developed the flu. [5pts]

d. Provide an interpretation of your interval in the context of the data. [2pts]

e. According to your confidence interval in c), is it reasonable to assume that fewer than 1% of all people vaccinated will develop the flu? Clearly answer “yes” or “no”. _______. And provide an explanation for your answer. [2pts]

Answer 1

Solution :

a) For the given scenario the most appropriate statisic will be sample proportion. The sample proportion of subjects who were vaccinated with the vaccine but develoed the flu is given by,

$\large \hat{p} = \frac{32}{4200} = 0.0076$

The observed value of the statistic is p̂ = 0.0076.

b) We can use normal distribution to construct a confidence interval if np̂ ≥ 10 and n(1 - p̂) ≥ 10.

We have, n = 4200 and  p̂ = 0.0076

np̂ = 31 which is greater than 10.

n(1 - p̂) = 4168 which is greater than 10.

Hence, sample size is large enough to use the normal distribution.

c) The 95% confidence interval for population proportion is given as follows :

$\large \left ( \hat{p} \pm Z_{\frac{0.05}{2}}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right )$

Where, p̂ is sample proportion, n is sample size and Z_(0.05/2) is critical z-value to construct 95% confidence interval.

We have,  p̂ = 0.0076, n = 4200

Using the calculator we get, Z_(0.05/2) = 1.96

Hence, the 95% confidence interval for the proportion of the population that were vaccinated with the vaccine but still developed the flu is,

0.0076 + 1.96 x 0.0076 x (1 – 0.0076) 4200

= (0.00760.0026)

$\large =\left (0.0076 - 0.0026, 0.0076 + 0.0026 \right )$

$\large =\left (0.0050, 0.0102 \right )$

The 95% confidence interval for the proportion of the population that were vaccinated with the vaccine but still developed the flu is (0.0050, 0.0102).

d) Interpretation : We are 95% confident that the true proportion of the population that were vaccinated with the vaccine but still developed the flu lies between 0.0050 and 0.0102.

e) Since, the 95% confidence interval also contain the value greater than 1% = 0.01, there it is not reasonable to assume that fewer than 1% of all people vaccinated will develop the flu.