What I need the most is part A and also learn how to interpret part B when I gen

Question

What I need the most is part A and also learn how to interpret part B when I generate the random numbers.

A candidate believes she is favored by 60% of the voters. If this is the case, use an a) propriate t to find how likely is it that a random sample of 25 voters would show fewer than 50% favoring the candidate? b) Using Excel's random number generator (see section 13.5.2), generate the outcome of 1000 simulations of the experiment described in a i e 1000 values of a binomial random variable with p .60 and 25. Determine the fraction of these experiments in which the number of successes was less than 50%. Compare with the theoretical probability found in a)

Explanation / Answer

SOLUTION

Let X = Number of voters in the sample of 25 favouring the candidate. Then, X ~ B(n, p), where n = 25 and p = 0.6 [0.6 is inferred from ‘candidate believes …….. If this is the case’]

Part (a)

Likely-hood that a random sample of 25 voters would show fewer than 50% favouring the candidate = P(X < 12.5) = P(X = 0) + P(X = 1) + …… + P(X = 12) = 0.153768 ANSWER

The above probability is directly obtained using Excel Function.

Part (b)   

How to interpret simulation results

When simulation is done, each run would produce 25 integers lying from 0 to 25, both inclusive. For each of these runs, count the number of integers that are less than or equal to 12, let us call this number as n1, n2, ………, n1000. Divide each ni by 25 to get the proportion, let us call it as pi. Then, find the average of these pi’s, i.e., (p1+ p2 + p3 + …… + p1000)/1000. Finally, compare it with 0.1537 68 as obtained in Part (a). It will be pretty close. DONE    

[This exercise has possibly been given to visualize that probability is the limiting value of sample proportion as the number of samples gets larger and larger. It would be an interesting and enlightening exercise to plot cumulative proportions at 50, 100, 150, 200, ……., 1000. The graph would be seen to be erratic initially, but gradually it would hover around 0.153768 very closely.]

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