Answer to Question #227840 in Statistics and Probability for Carol

Question

Which of the following statements are true and which are false? Justify your answers!

(a) The population variance is always larger than the standard error of the mean.

(b) The sample mean of a random variable X has normal distribution if X is normal, and is

approximately normal even if X does not have the normal distribution, as long as the sample

size is large enough.

(c) The sample proportion in a random sample has a binomial distribution, but can often be

approximated by the normal distribution.

(d) The standard error of the proportion will increase when the sample size gets larger.

(e) A point estimator is better than an interval estimator because it is more precise.

(f) The sample mean is an unbiased estimator of the population mean but it is not always con-

sistent, if the population variance is very large.

(g) Assume that we are estimating the population mean with confidence level 1 

Answer 1

Solution:

(a) True - the standard error is the ratio of the population standard deviation and the sample size hence the standard error will always be less than the population variance.

(b) The given statement is true because according to the central limit theorem, the sampling distribution of the sample mean is always normally distributed irrespective of the distribution of the population as long as the sample size is sufficiently large.

(c) The given statement is true because the favorable proportion is considered constant for each sample point and with a fixed sample size it is always considered binomially distributed. The binomial distribution can be approximated to normal if the sample size is large enough.

(d)

The given statement is false because the standard error is inversely proportional to the proportion standard deviation due to which the standard error is smaller for larger data.

$SE_p=\sqrt{\dfrac{pq}{n}}$

(e)

The given statement is false because a point statement is never a more precise estimate as compared to an interval. it can be easily understood that an interval includes values around the point estimate which are also likely like the point estimate and hence add to the confidence level of the interval while on the other hand, the point estimate is a single value contributing to the confidence level.

(f)

The given statement is false because the sample mean is actually an unbiased estimator of the population mean irrespective of the variance of the data.