Question

One representative from the CDC claimed that as many as 25% of people who have Covid-19 may not be showing any symptoms. If this is true, what is the probability of randomly choosing 450 people who are experiencing no symptoms and 110 or less of them actually have the virus?

Make sketches and label all variables and give the appropriate calculator command.

What is the inequality that must be true in order to treat the sample proportions in this problem as being normally distributed?

Answer 1

This is a binomial event where the successful event is having the virus

p = 25% which independent and same for all the subjects

n = 450

np = 112.5

Here as the sample size n > 30 we can use the normal approximation also np> 10

These are the formulas for the binomial expected value and Std dev.

z-score = (x - 112.5) / 9.1856

We need to find the probability that  110 or less of them actually have the virus

But since binomial is a discrete distribution and we are approximating using a continuous normal distribution we will need to do a continuity correction.

P(X 110) = P( X < 110+0.5)

= P(X < 110.5)

........if we use X < 110, that means X = 109 , 108 nd so on but it is said that X = 110, 109. We can not write X < 109.5 because again it will mean X = 109 , 108, so we take a value a little greater than 110, so we can use it with the sign '<'. Therefore we take 110.5

= P( Z < -0.22)

= 1 - P( Z < 0.22)

= 1 - 0.5862

P(X 110) = 0.41382