Define the estimate of a population parameter.
To obtain information about a population parameter, such as the value of population mean µ or population proportion π, a random sample of objects from the population is usually created. Then, the sample statistic found from the values of the sample observations is used for predictions about population parameter.
Define the point estimation.
Generally, the process of estimating a population parameter by a single number derived from the sample is called point estimation.
What are the three basic features of a point estimation?
Define sampling distribution.
Generally, the probability distribution of a sample statistic is called sampling distribution of the statistic. A sample statistic is a random variable before the sample observation’s selection. However, after the sampling, a statistic is simply a number.
An investigator has found the following sample values for a continuous variable: 8,12,18, and 24. Calculate the point estimate of the population arithmetic mean?
It would be '15.5'.
If you create one sample from a population and calculate a statistic about a population parameter from this sample, how do we call this estimation method?
We call this 'Point Estimation'.
A teacher has counted the students in a class for a course in the morning and have noticed that 10 children is absent out of 67 students. The teacher has counted that there are 41 female students in the course this morning. What is the population proportion estimate of female students in this class?
The population proportion estimate of female students in this class is 41/57.
When you use sample arithmetic mean to estimate the population mean, if the sample arithmetic mean of a variable is 65, then what is the point estimate of the population arithmetic mean of the same variable?
The point estimate of the population arithmetic mean of the same variable is 65.
The frequency of broken products in a sample of 4000 is 80. What is the point estimate of population proportion for broken products?
The point estimate of population proportion for broken products is '0.02'.
In a population there are 120 observations. An investigator has measured the weight of 12 objects and the results are: 12, 23, 14, 14, 15, 13, 10, 10, 9, 10, 9, and 11. Using the point estimation, what is the value of the population arithmetic mean weight?
Using the point estimation approach, we can safely say that the value of the population weight is equal to the 12.50 kilogram.
What is the sampling distribution of the sample mean?
The sampling distribution of the sample mean is the probability distribution of all possible sample means that can be created from a population with a fixed sample size of n.
Define the central limit theorem.
Central limit theorem is one of the most important theorems in statistics. If you take independent samples one by one from a population with a known arithmetic mean and standard deviation and calculate the means of these samples, central limit theorem states that the shape of the distribution of the sample means will be approximately Normal distribution. Central Limit Theorem also indicates that without the regard of the shape of a population parameter’s distribution (symmetric, asymmetric, left skewed or right skewed), the sampling distribution of the sample mean approximates to the normal distribution as the sample size increases.
In an investigation there are 5 observations. The values the values of the observations for some variable are found as 18, 20, 16, 24, and 22. Please, create the sampling distribution of the sample mean for size n=2 on a table environment.
First of all, lets associate each observation with a letter as follows: A=18, B=20, C=16, D=24, E=22. Sample size is given as n = 2. Now sampling distribution of the sample mean can be shown using in the following table.
Using the sample means, it is easy to create a bar chart of the arithmetic means as follows.
For what purpose is interval estimation used?
While it is known that the value of the point estimate of a population parameter gets closer to real value of population parameter as sample size increases, we would like to measure how close the point estimate really is to the actual value. For this purpose, interval estimation is used. An interval estimation of a population parameter, demonstrated by θ in general, includes two limits or boundaries.
What are the two different situations in calculating Confidence Interval for Population Mean µ?
There are two important situations in the calculation of confidence interval for a population mean. The first case is that we use sample data to estimate the unknown value of population mean µ with the value of sample mean found from a random sample of size n, and also the population standard deviation σ is known beforehand. The second case is that we use sample data to estimate the unknown value of population µ with the value of sample mean found from a random sample of size n, and also the population standard deviation σ is not known. In the second case, instead of the population standard deviation σ, the sample standard deviation s is used as the substitute.
In an interval estimation problem, the sample arithmetic mean is 45 and the lower limit of the confidence interval is 35, then, for this problem, what is the upper limit of the confidence interval?
The upper limit of the confidence interval is 55.
In an interval estimation problem, the investigator has found the limits of the confidence interval as (140, 170) when the significance level is 0.01. What is the sample arithmetic mean that was used in this confidence interval problem?
The sample arithmetic mean that was used in this confidence interval problem is 155.
In order to estimate the population parameter if the investigator creates a range, what is this estimation procedure called in statistics?
This estimation procedure is called Interval Estimation.
In an interval estimation problem, the population standard deviation is not known, and the sample standard deviation is found as 48, if there are 12 observations in the sample, what is the value of the standard error that is be used in the confidence interval calculations?
The value of the standard error that is be used in the confidence interval calculations is 4.
In an investigation, sample proportion is found as 0.42. There are 60 observations in this sample. If you want to find the 95% confidence interval for the population proportion, what is the values of zα/2 and standard error in the confidence interval formula?
zα/2 = z0.05/2 is what we are looking for, and the value of the z0.05/2 from the standard normal distribution table is 1.96. Now the standard error in the confidence interval formula is as follows:
