Question

We use sample data to estimate population parameters and point estimate and interval estimate are two typical tools of such estimation.

Please discuss them answering the following questions.

1) Briefly discuss about how point estimate and confidence interval are different to each other.

2) Discuss about way(s) to secure unbiased and efficient point estimators.

3) Discuss about the effects of confidence level (or, Z-score) and sample size on the width of confidence interval. Is it good or bad to have wide/narrow confidence interval?

Answer 1

a. Point estimation gives us a particular value or a single value as an estimate of the population parameter like mu Sigma etc while  Interval estimation gives us a range of values which is likely to contain the population parameter. This interval is called a confidence interval. That is the population parameters lies in the confidence interval.

b. An unbiased estimator is an accurate statistic that’s used to approximate a population parameter. “Accurate” in this sense means that it’s neither an overestimate nor an underestimate. If an overestimate or underestimate does happen, the mean of the difference is called a “bias.”

We can obtain unbiased estimators by avoiding bias during sampling and data collection.

It is desirable for a point estimate to be:

(1) Consistent : The larger the sample size, the more accurate the estimate.

(2) Unbiased. The expectation of the observed values of many samples (“average observation value”) equals the corresponding population parameter. For example, the sample mean is an unbiased estimator for the population mean.

(3) Most efficient or best unbiased—of all consistent, unbiased estimates, the one possessing the smallest variance (a measure of the amount of dispersion away from the estimate). In other words, the estimator that varies least from sample to sample. This generally depends on the particular distribution of the population. For example, the mean is more efficient than the median (middle value) for the normal distribution but not for more “skewed” (asymmetrical) distributions.

c. Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error.

Increasing the confidence level increases the error bound, making the confidence interval wider.

Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

A narrow confidence interval implies that there is a smaller chance of obtaining an observation within that interval, therefore, our accuracy is higher.

Also a 95% confidence interval is narrower than a 99% confidence interval which is wider.

The 99% confidence interval is more accurate than the 95%.

It is totally depends on the Problems, we can't say that which one is good.

But generally we consider 95% confidence level.

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