Answer:
[tex]Range = 85[/tex]
[tex]\sigma = 28.71[/tex]
[tex]Interval = [666.78, 781.62][/tex]
Step-by-step explanation:
Given
The data for 25 undergraduates
Solving (a): Range and Standard deviation
The range is:
[tex]Range = Highest - Least[/tex]
From the dataset:
[tex]Highest = 772[/tex]
[tex]Least = 687[/tex]
So:
[tex]Range = Highest - Least[/tex]
[tex]Range = 772-687[/tex]
[tex]Range = 85[/tex]
The standard deviation is:
[tex]\sigma = \sqrt{\frac{\sum(x - \bar x)^2}{n}}[/tex]
First, calculate the mean
[tex]\bar x = \frac{769 +691 +............+715}{25}[/tex]
[tex]\bar x = \frac{18105}{25}[/tex]
[tex]\bar x = 724.2[/tex]
So, the standard deviation is:
[tex]\sigma = \sqrt{\frac{(769-724.2)^2 +(691-724.2)^2 +(699-724.2)^2 +(730-724.2)^2 +............+(715-724.2)^2}{25}}[/tex]
[tex]\sigma = \sqrt{\frac{20604}{25}}[/tex]
[tex]\sigma = \sqrt{824.16}[/tex]
[tex]\sigma = 28.71[/tex]
Solving (b): The interval of the 95% of the observation.
Using the emperical rule, we have:
[tex]Interval = [\bar x - 2*\sigma, \bar x+ 2*\sigma][/tex]
[tex]Interval = [724.2 - 2*28.71, 724.2 + 2*28.71][/tex]
[tex]Interval = [666.78, 781.62][/tex]
