Question

Asked Feb 27, 2019

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In a certain community, it is accepted that 85% of households are two-income households. You select a random sample of 150 households. What is the probability that more than 130 of them are two income households?

Step 1

**Binomial distribution:**

A random variable is said to follow binomial distribution if it satisfies the criteria given below:

- The number of observations or trails must be fixed.
- Each observation or trail must be independent.
- The probability of success must be same for all the trails.

The pmf for the binomial distribution or the probability of obtaining *x* successes in *n* independent trails of a binomial experiment is,

*P*(*x*) = *nCx* * *p ^{x} * *(1 –

Here, it is given that, 85% of households are two-income households.

The probability of success (Households that are two-income) is *p *= 85/100 = 0.85.

The probability of failure (Households that are not two-income) is 1 – *p *= 0.15.

The sample number of households selected is *n *= 150 (Which is large in size).

The variable *x *be the number of households that are two income.

- Here, the number of households selected is
*n*= 150 is fixed. - Each of the selected 150 households are independent of each other.
- The probability of success (Households that are two-income) is
*p*= 0.85 (Same for all 150 households).

Hence, the variable *x *= number of households that are two income is a binomial random variate.

Step 2

**Normal approximation to binomial distribution for large sample size:**

The mean and variance for the binomial variate is *E*(*x*) = *n ** *p *and *V*(*x*) = *n ***p **(1 – *p*).

Here, *p *= 0.85, 1 – *p *= 0.15 and *n *= 150.

Since, the sample size is large, binomial variate can be transformed to normal variate.

The mean and variance of the variable *x *(number of households that are two income) is *E*(*x*) = *n ** *p *= 150*0.85 = 127.5 and *V*(*x*) = *n ***p **(1 – *p*) = 150*0.85*0.15 = 19.125.

If *X* follows binomial distribution with parameters *np *and *npq, *the binomial variate tends to normal as follows:

Step 3

**Find the probability that more than 130 households are two-income:**

Here, the variable number of households that are two-income is more of 130. That is, *x *> 130.

Th...

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