An article in journal of american statistical association (1990,vol. 85, pp. 972-985) measured weight of 30 rats under experimentcontrols. suppose that there are 12 underweight rats.

a) calculate a 95% two-sided confidence interval on the trueproportion of rats that would show underweight from theexperiment.

b) using the point estimate of p obtained from the preliminarysample, what sample size is needed to be 95% confident that theerror in estimating true value of p is less than 0.02?

a) [6.741, 17.259]

b) 2,305

Step-by-step explanation:

a)

Let p be the proportion of underweight rats and n the sample size.

n=30

p= 12/30 = 0.4

Since, both np>10 and n(1-p)>10 and the sample size = 30 is big enough, we can approximate the distribution with a Normal curve of mean 12 and standard deviation

We then find the values a, b such that the area of the curve between [a, b] equals 95%=0.95.

That values can be easily worked out with a spreadsheet and we have the 95% confidence interval [6.741, 17.259]

See picture attached

b)

Using Simple Random Sampling in an infinite population (this is such a large population that we do not know the exact number) we have that the sample size should be the nearest integer to

where

Z= the z-score corresponding to the confidence level, in this case 95%, so Z=1.96 (this means that the area under the Normal N(0,1) between [-1.96,1.96] is 95%=0.95)

p= the proportion of underweight rats = 0.4

q = 1-p = 0.6

e = the error proportion = 0.02

Crunching the numbers

The sample size should be at least 2,305.

∠cbe and ∠deb are same-side interior angles.

i took the test