Question

The mean per capita income is 21,604 dollars per annum with a standard deviation of 727 dollars per annum. What is the probability that the sample mean would differ from the true mean by less than 36 dollars if a sample of 193 persons is randomly selected

Answer 1

The probability that the sample mean would differ from the true mean by less than 36 dollars is 0.7549.

Explanation:

According to the Central Limit Theorem if an unknown population is selected with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from this population with replacement, then the distribution of the sample means will be approximately normally.

Then, the mean of the sample means is given by,

`\mu_(\bar x)=\mu`

And the standard deviation of the sample means is given by,

`\sigma_(\bar x)=(\sigma)/(√(n))`

The information provided is:

`\mu=21604\\\sigma=727\\n=193`

As the sample size is large, i.e. n = 193 > 30 the central limit theorem can be used to approximate the sampling distribution of sample mean per capita income.

Compute the probability that the sample mean would differ from the true mean by less than 36 dollars as follows:

`P(\bar X-\mu<36)=P((\bar X-\mu)/(\sigma_(\bar x))<(36)/(727/√(193)))\\\\=P(Z<0.69)\\\\=0.7549`

Thus, the probability that the sample mean would differ from the true mean by less than 36 dollars is 0.7549.