Question

a recent poll suggests that 67% of canadians use their mobile device to comparison shop while they are in a store. you are to randomly pick 4 canadians from a population made up of people who have a mobile device. you ask each person the question: do you use your mobile device to comparison shop while you are in a store? the number who respond \yes\ is counted and represented by the random variable x. its probability distribution table is given below

(a) what can you say about the tendency of x? select all that apply.
a. x is more likely to take on larger rather than smaller values.
b. the probability distribution of x is perfectly symmetric.
c. the probability distribution of x is right - skewed.
d. the probability distribution of x is left - skewed.
e. x is more likely to take on smaller rather than larger values.

(b) find the expected value of x.
e(x)=(use two decimals in your answer)

(c) find the standard deviation of x.
sd(x)=(use two decimals in your answer)

Explanation:

Step1: Analyze tendency of \(X\)

• Compare probabilities: \(P(X = 0)=0.0119\), \(P(X = 1)=0.0963\), \(P(X = 2)=0.2933\), \(P(X = 3)=0.397\), \(P(X = 4)=0.2015\). As \(x\) increases from \(0\) to \(3\), \(P(X=x)\) increases and then decreases. Also, \(P(X = 3)+P(X = 4)=0.397 + 0.2015=0.5985\) and \(P(X = 0)+P(X = 1)=0.0119+0.0963 = 0.1082\). So \(X\) is more likely to take on larger rather than smaller values.
• Check symmetry: A symmetric distribution has \(P(X=a)=P(X = n - a)\) for a discrete distribution over \(n\) values. Here \(n = 4\), \(P(X = 0)

eq P(X = 4)\) and \(P(X = 1)
eq P(X = 3)\), so it's not symmetric.

• Check skewness: Since the tail is on the left (smaller \(x\) values have smaller probabilities compared to the right - side for higher \(x\) values), the distribution is left - skewed.

Step2: Calculate expected value \(E(X)\)

The formula for the expected value of a discrete random variable is \(E(X)=\sum_{x}x\cdot P(X = x)\).

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Step3: Calculate \(E(X^{2})\)

The formula is \(E(X^{2})=\sum_{x}x^{2}\cdot P(X = x)\)

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Step4: Calculate variance \(V(X)\)

The formula for variance is \(V(X)=E(X^{2})-(E(X))^{2}\)

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Step5: Calculate standard deviation \(SD(X)\)

The formula for standard deviation is \(SD(X)=\sqrt{V(X)}\)

$$SD(X)=\sqrt{0.8841}\approx0.94$$

Answer:

(a) A. \(X\) is more likely to take on larger rather than smaller values; D. The probability distribution of \(X\) is left - skewed.
(b) \(2.68\)
(c) \(0.94\)