Solved Questions on Parameters and Statistics
Q1. Suppose we want to find the average height of the students from all over the country. We have sample of size 50. The average height of the sample is 160 cm. The sample standard deviation is 4. Construct a 95% confidence interval for population mean
Solution:
Sample mean=160 cm
Number of samples=50
Sample Standard Deviation = 4
The formula for the 95% confidence interval for the population mean is
[Tex]\bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \\=160\pm1.96\times(\frac{4}{\sqrt50}) \\=(158.89,161.11)[/Tex]
So the average height of the population ranges between 158.89 cm and 161.11 cm
Q2. Calculate sample variance of the data 40,20,30,50,20
Solution:
To calculate the sample variance first we need to find sample mean
[Tex]\bar{x}=\frac{(40+20+30+50+20)}{5} \\=32[/Tex]
Now we will calculate the sample variance by finding the difference between the datapoint and the sample mean, then perform the sum of squared differences and then divide by n-1 where n is the number of samples
[Tex](40-32)^2=64 \\(20-32)^2=144 \\(30-32)^2=4 \\(50-32)^2=324 \\(20-32)^2=144[/Tex]
Let sum of differences be denoted by m
Sum of differences = 64+144+4+324+144=680
Sample variance = [Tex]\frac{m}{n-1} \\=\frac{680}{5-1} \\=170[/Tex]
Q3. Now calculate the standard deviation of the above data
Solution:
By definition we know that standard deviation is the square root of the variance.
So [Tex]s=\sqrt{170} \\=13.0384 \\\approx13.04[/Tex]
Q4. Calculate sample mean of the values 63,65,62,67.
Solution:
The sample size is 4
The sample values are 63,65,62,67
The sample mean is
[Tex]\bar{x}=\frac{(63+65+62+67)}{4} \\=\frac{257}{4} \\=64.25[/Tex]
Parameters and Statistics
Statistics and parameters are two fundamental concepts in statistical theory. Although they may sound equal, there is a sharp difference between the two. One is used to represent the population, and the other is used to represent the sample. Now we will focus on the sample and population:
Population: A population refers to the whole data. It is the dataset that the statisticians use to derive conclusions or gain insights about the data.
Sample: Sample refers to the small dataset. It is considered to be a subset of population. Since population can be huge and may be difficult to examine, Statisticians usually consider a subset or sample, perform Statistical analysis and derive conclusions about the Population.
It is to be noted that the sample to be selected should be random in nature. If the subgroup or sample is not randomly selected, it may produce biased results.
Table of Content
- Parameters
- Statistics
- Relationship Between Sample and Population
- How to derive Population Parameter using Statistics?
- Types of Parameters and Statistics
- Difference between Parameters and Statistics
- Solved Questions on Parameters and Statistics
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