Do vaccines have side effects which make the recipients ill? In most vaccine trials some recipients receive the actual vaccine while some receive a placebo, basically a fake vaccine that has no real effect. However, the recipients do not know whether they received the vaccine or the placebo. Neither do the vaccinators/physicians. That why this is sometimes called a double-blind test. To test whether the vaccine has side effects we would like to examine the proportion of vaccine vaccinated recipients who have side effects,p_v

, and compare this to the proportion of placebo vaccinated recipients who have side effects,p_p

. We are going to examine this question by randomly selecting NV=1203 vaccine vaccinated recipients and NP=970 placebo vaccinated recipients. Let XV equal the number of vaccine vaccinated recipients in our sample who report side effects. Let XP equal the number of placebo vaccinated recipients in our sample who report side effects. Suppose XV = 632 and XP = 432. Let pvhat be the sample proportion of vaccine vaccinated recipients in our sample who report side effects Let pphat be the sample proportion of placebo vaccinated recipients in our sample who report side effects. Letp_v

be the (unknown) true proportion of vaccine vaccinated recipients who report side effects. Letp_p

be the (unknown) true proportion of placebo vaccinated recipients who report side effects

a)Calculate an unbiased point estimate ofp_v

.

b) We wish to construct a 95 % classical confidence interval forp_v

. What is the critical value multiplier zstar?

c) Create a 95% classical confidence interval forp_v

?,

d) How long is the 95% classical confidence interval forp_v

?

e) In terms ofp_v

= p and NV = n, give the formula for the standard deviation of the distribution of the sample proportion pvhat.(R code)

sqrt(p*(1-p)/n)

n*p*(1-p)

sqrt(n*p*(1-p))

p*(1-p)/n

f) Calculate an unbiased point estimate ofp_p

g) Calculate an unbiased point estimate ofp_v-p_p

h)Based on this data, calculate a 95% classical confidence interval forp_v-p_p

.,

i. How long is the 95% classical confidence interval forp_v-p_p

calculated above?

j) Copy your R script for the above into the text box here.

7.

DETAILSMY NOTES

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The blood pressures of 18 randomly chosen 65-year-old Covid-19 patients were measured. The following are the measurements:

136.00, 134.67, 149.72, 151.68, 125.20, 127.18, 119.61, 132.70, 147.82, 138.26, 150.98, 129.16, 137.45, 177.52, 156.19 ,122.73 ,122.17, 158.74

Research has shown that blood pressures are normally distributed. So, we assume that our sample comes from a normal population with an unknown mean of?and an unknown standard deviation of?. We would like to test whether the average blood pressure of 65-year-old Covid-19 patients is greater than 132.

The null hypothesis is thusH₀:?=132

. We will test this against the alternativeH_a

.

We want to test at the 4% level.

Letx= the sample mean and s = the sample standard deviation.

a) What should the alternative hypothesis,H_a

, be?

H_a:?132

H_a:?<132

H_a:?=132

H_a:?=3%

H_a:?>132

b) What is the formula for your test statistic?

T =x-132

s

19

T =x-132

s

18

T =x-132

s

18

T =x-132

s

17

T =x-3%

s

c) What value does your test statistic,T, take on with the sample data?

d) What type of probability distribution does your test statistic,T, have?

normal

binomial

t

Chi-Squared

Cauchy

e) How many degrees of freedom does T have?

f) Calculate the critical value, tstar, for your test.(positive value)

g) For what values of your test statistic, T, is the null hypothesis rejected?

|T - tstar| < .04

T > tstar/2 or T < -tstar/2

T < tstar

T > tstar

T > tstar or T < -tstar

h) Calculate the p-value for this test.

i) Is the null hypothesis rejected? (Y/N)

N

Y

j) If we ran 300, 4% level tests then how many times would we make a Type I error?

k) Create a 96% confidence interval for the average blood pressure of 65-year-old Covid-19 patients based upon the above data. (,

)

l) Copy your R script or any other comments for the above into the text box here.