Estimation

In a roadside interview survey, a random sample of all vehicles crossing each of a set of intersections is taken. Assume that every trip made in the area of interest over the duration of the study is equally likely to be sampled by this method. One question asked in survey concerns the total number of home-based round trips made by all the members of the household to which the sampled tripmaker belongs. The frequency distribution of the response to this question is:

Number of household trips Number of respondents
1 700
2 1500
3 800
4 600
5 400

a) Explain why the relative frequency of distribution for the sample does not provide a correct estimate of the distribution of households trip frequency that one would have obtained from a random sample of households.
b) Assume that it is known from some other source that E(T), the average number of trips made by the household in the area of interest is equal to 1.85, and that the standard deviation of T is equal to 1.2. Derive an expression in terms of E(T) and V(T) for E(W), the expected number of household trips reported by the respondents sampled in the roadside survey, and calculate E(W) using the expression. Verify that your answer is equal to the sample mean calculated directly from the sample information. (V(T) is variance of T)

c) What is the probability that a randomly selected household in the study area makes no trip at all?