I survey a simple random sample of 1000 US households and fi

I survey a simple random sample of 1000 U.S. households and find out their income.

(a) True or false: By the Central Limit Theorem, the incomes in the population will have an approximately normal distribution.

(b) True or false: By the Central Limit Theorem, the incomes in the sample will have an approximately normal distribution.

Solution

d) True - The Central limit theorem states that , the distribution of the sample mean approaches the normal distribution for sufficiently larger samples (n>=30) , even if the parent distribution is not normal. The population is very large. No correction is needed in the calculation of variance. Therefore by central limit theorem, the incomes in the population will have an approximately normal distribution.

e) True - The central limit thorem states that the distribution of the sample mean approaches the normal distribution for sufficiently larger samples(n>=30).Here, n = 1000. Therefore by central limit theorem the incomes in the sample will have an approximately normal distribution.

If sample of size n are drawn at random from any population with a finite mean and standard deviation then the sampling distribution of the sample mean approximates a normal distrbution as n increases. The mean of this sampling distribution approximates the population mean.