Checking smartphone. A 2014 Bank of America survey of U.S. adults who own smartp

Question

Checking smartphone. A 2014 Bank of America survey of U.S. adults who own smartphones found that 35% of the respondents check their phones at least once an hour for each hour during the waking hours.6 Such smartphone owners are classified as "constant checkers." Suppose you were to draw a random sample of 10 smartphone owners. The number in your sample who are constant checkers has a binomial distribution. What are n and p? Use the binomial formula to find the probability that exactly two of the 10 are constant checkers in your sample. Use the binomial formula to find the probability that two or fewer are constant checkers in your sample. What is the mean number of owners in such samples who are constant checkers? What is the standard deviation?

Explanation / Answer

Solution:

The binomial distribution formula is:

b(x; n, P) = nCx * Px * (1 – P)n – x

Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials

(a)here n=10 smartphone owners.

p=35%=35/100=0.35

q=1-p=1-0.35=0.65

le X denote the number in the sample with constant checkers.

to find probability P(X=2)

since 2 out of 10 are constant checkers in sample.

P(exactly 2 out of 10)=

P(X=2)=10 C2 (0.35)2 (0.65)10-2

   = 10.9/2.1(0.35)2 (0.65)8

=0.1756

the prob that exactly 2 out of 10 are constannt checkers is 0.1756

(c)to find the probability

P(2 or fewer out of 10)=

P(X<=2)=P(X=0)+P(X=1)+P(X=2)

=10c0(0.35)0(0.65)10-0+10C1(0.35)1(0.65)10-1 +0.1756

=0.6510 +10(0.35)(0.65)9 +0.1756

=0.26155

(d)mean is given by np

mean =np=10(0.35)=35

standard deviation =sqrt(npq)=sqrt(10*0.35*0.65)=sqrt(2.275)=1.5083

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