Question

Do you want nuclear power plant in Manhattan? Out of a random sample of 289 Manhattan residents, 105 residents were in favor of a nuclear power plant in Manhattan. Let p be the (unknown) true proportion of Manhattan residents who are in favor of a nuclear power plant in Manhattan. Let phat be the sample proportion.
a)Obtain an unbiased point estimate of p.
b) We wish to construct a 97 % classical confidence interval for p. What is the critical value multiplier zstar?
c) Create a 97% classical confidence interval for p? ,
d) How long is the 97% classical confidence interval for p?
e) In terms of p and n, give the formula for the standard deviation of the distribution of the sample proportion phat.(R code) n*p*(1-p) sqrt(p*(1-p)/n) p*(1-p)/n sqrt(n*p*(1-p))
f) Based on this data, estimate the standard deviation of the proportion of Manhattan residents who are in favor of a nuclear power plant in Manhattan. (4 digits of accuracy)
g)Assuming the same value of the sample proportion what is the smallest sample size for which the length of the 97% confidence interval for p is less than or equal to .09?
h) Show your R script for the above.

Answer 1

a) The unbiased estimate is the sample proportion computed here as:

p = 105/ 289 = 0.3633

b) From standard normal tables we get:

P( -2.17 < Z < 2.17 ) = 0.97

Therefore the critical z value here would be 2.17

c) The 97% confidence interval here is computed as:

This is the required confidence interval here.

d) The length of the interval here is computed as:

e) The standard deviation of the distribution of the sample proportion is computed as:

f) The standard deviation is computed here as:

g) The length of interval here is 0.09, therefore MOE = 0.09/2 = 0.045

The MOE is computed here as:

Therefore 538 is the minimum sample size required here.