In 2010. a survey of 400 homes in a region found that 200 had overestimated market values.

The sample statistics are used to estimate the population estimates, in this case the population parameter to be estimated in the population proportion 'pi'. 

The number of successes in the sample are assumed to estiimate the number of successes in the population given the current sample size. 

To estimate the population proportion, the number of successes in the sample (200) is divided by the sample siz (400).  Hence the estimate of the populaton proportion "pi" is 200/400 = 0.5

b. The standard normal distribution is used when constructing the confidence interval for such problems because it is assumed that the occurences of the events follows a normal distribution. The sample is also large enough for the application of the normal distribution to be considered valid. 

c. 

The confidence interval is constracted using the standard normal distribution table, and the formula stated below is applied. 

confidence interval for estimating the population proportion is p_hatch +/- Z_score * SE, the standard error for this confidence interval is 

standard error for proportion =  sqrt(p_hatch * (1- p_hatch)/n) ;- sqrt(0.5 * (1 - 0.5)/400) = 0.025

The confidence interval  = 0.5 +/- 1.96 * 0.025  = (0.5 +/- 0.049), Hence the confidence interval is (0.451, 0.549)

Interpretation of the confidence interval must follow the defined population.  We are 95% confidence that the percentage of the people who overestimated the market value is between (45.1%, 54.9%). 

The claim that the proportion is 0.37 is not valid because this value is not included in this interval.