Answer:
The mean and standard deviation of random samples to the population parameters is less than the mean and standard deviation of population.
Step-by-step explanation:
Given : For a population, the mean is 19.4 and the standard deviation is 5.8.
The following random samples to the population parameters.
12, 15, 17, 20, 13, 11, 18, 19, 15, 14
To find : Compare the mean and standard deviation ?
Solution :
1) We find the mean of the random sample,
[tex]M=\frac{\sum x_n}{n}[/tex]
[tex]M=\frac{12+15+17+20+13+11+18+19+15+14}{10}[/tex]
[tex]M=\frac{154}{10}[/tex]
[tex]M=15.4[/tex]
The Mean of random sample 15.4 is less than 19.4 the mean of population.
2) We find the standard deviation of the random sample,
[tex]s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x-\overline{x})^{2}}[/tex]
[tex]s=\sqrt{\frac{1}{10-1}\sum_{i=1}^{n}(x-\overline{x})^{2}}[/tex]
Solving summation separately,
[tex]\sum_{i=1}^{n}(x-\overline{x})^{2}=(12-15.4)^2+(15-15.4)^2+(17-15.4)^2+(20-15.4)^2+(13-15.4)^2+(11-15.4)^2+(18-15.4)^2+(19-15.4)^2+(15-15.4)^2+(14-15.4)^2[/tex]
[tex]\sum_{i=1}^{n}(x-\overline{x})^{2}=82.4[/tex]
Substitute back,
[tex]s=\sqrt{\frac{1}{9}(82.4)}[/tex]
[tex]s=\sqrt{9.155}[/tex]
[tex]s=3.025[/tex]
The Standard deviation of random sample 3.025 is less than 5.8 the standard deviation of population.
