A sample of 25 undergraduates reported the following dollar amounts of entertainment expenses last year:

769 691 699 730 711 765 702 718 719 712 768 688 757 695 768 735 709 758 708 693 736 700 687 772 715

Required:
a. What are the range and standard deviation?
b. Use the Empirical Rule to establish an interval which includes about 95 percent of the observations.

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]

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