B) What is the probability that less than three households withdrew funds from a

Question

B) What is the probability that less than three households withdrew funds from a retirement account for needs other than retirement?
C) What is the probability that more than four households withdrew funds from a retirement account for needs other than retirement?
D) What are the Mean and Standard Deviation for this distribution?
A budgeting Web site reported that 25% of US households have withdrawn money from a 401(k random sample of 11 U.S. households was selected. Complete parts a through e below or other retirement account tr needs other than ret en tin 2011

Explanation / Answer

Let X = number of households in a sample of 11 US households which withdrew funds from a retirement account for needs other than retirement.

Then, X ~ B(11, p), where p = probability that a household withdrew funds from a retirement account for needs other than retirement, which is estimated by the proportion of households which withdrew funds from a retirement account for needs other than retirement.

‘A budgeting Web site reported that 25% of US households have withdrawn money from a 401(k)’ => p = 0.25.

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then

probability mass function (pmf) of X is given by

p(x) = P(X = x) = (nCx)(px)(1 - p)n – x, x = 0, 1, 2, ……. , n ………………………………..(1)

[The above probability can also be directly obtained using Excel Function of Binomial Distribution] ……………………………………………………………………………….(1a)

Mean (average) of X = E(X) = np…………………………………………………………..(2)

Variance of X = V(X) = np(1 – p)…………………………………………………………..(3)

Standatd Deviation of X = SD(X) = { np(1 – p)} ………………………………………...(4)

Part (a)

Probability that exactly two households withdrew funds from a retirement account for needs other than retirement = P(X = 2)

= 0.2581 ANSWER

Part (b)

Probability that less than three households withdrew funds from a retirement account for needs other than retirement = P(X < 3)

= 0.4552 ANSWER

Part (c)

Probability that more than four households withdrew funds from a retirement account for needs other than retirement = P(X > 4)

= 1 – 0.8854

= 0.1146 ANSWER

Part (d)

Mean and Standard Deviation for this distribution

Vide (2) under Back-up Theory, Mean = (11 x 0.25) = 2.75 ANSWER 1

Vide (4) under Back-up Theory,

Standard Deviation = (11 x 0.25 x 0.75) = 2.0625 = 1.44 ANSWER 2

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