Question

It is believed that 52% of Americans like to eat at restaurants every week. You interview 7 Americans and ask if they like to eat at restaurants every week. The random variable x represents the number of Americans that like to eat at restaurants every week. 27a. Find the mean of the binomial distribution 276. Find the standard deviation of the binomial distribution.

Answer 1

TOPIC:Mean and standard deviation of Binomial distribution.

& Consider the I Theorem: following theorem as- If x ~ Bin (m, 6); then, E(X) = np. sd(x) = √ np (1-6). Proof - when x ~ Bin (m, p); then Pory of x P (x = x) = { (2) b* qu-x ; Jom, 220,1,2,n. o ; otherwise. where q = (1-P). So, E(X) in I X. P(x=x). 6x q n-x. n ! = { x 'n-x) "9" & nib* qu-x (x-D! (n-2)! = np. (n-1)! 6 x y q n-x (x-D! (n-x)! *a1 - . (*:!) *(0)-(a). a np (9+ By Binomial I Expansion formula anr. (1) h1 [:9 +6 27 Enp.

Again, E[x(x-7]. = *(x-7). P(x = 2). n- - ² x (x-). (2) b* qr-2 = 2(2-). Como (n.) ango = n(n-D p² (2) 2 * 2 g (2-2) - (x-2) x-2 n-x x = 2 = n(n-) #2 (a +p) ^-2. = n(n-D p2 (1) 0-2 =n(n-1 p² : 9+p=1) von (x) = [x(x-1) + E (x) - € 2(x). - (1-3 ^ + b - Phỏ. = np - np² a np (1-P). so, ( sd (x) = JMP(-6) (Proven).

= In the given question, we define, =) p = P (of an American like to eat) at restaurants every week, - 52 = 0.52. =) n a sample size = 7. Ket, x = No. of Americans in the sample, who like to eat at restaurants every week So, (xm Bin (n=7, p=0.52) 27. a) we need, =) mean (x). = E(X). = np. (By the theorem) .. = 7X0.52 . = 13.64 (Ang) 27.6 we need, =) Ad (x) = 1 np (1-6). (By the theorem 7X0.52X0.48. - 1.321817) CANO).