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Question Evaluate the formula χ2=(n−1)s2σ2\chi^{2}=\frac{(n-1) s^{2}}{\sigma^{2}} given σ=1.57,n=40,s=3.23\sigma=1.57, n=40, s=3.23. Round χ2\chi^{2} to three decimal places.

Studdy Solution

STEP 1

Assumptions
1. The formula given is χ2=(n−1)s2σ2\chi^{2}=\frac{(n-1) s^{2}}{\sigma^{2}}
2. The standard deviation (σ\sigma) is 1.57
3. The sample size (n) is 40
4. The sample standard deviation (s) is 3.23

STEP 2

First, we need to substitute the given values into the formula.
χ2=(n−1)s2σ2\chi^{2}=\frac{(n-1) s^{2}}{\sigma^{2}}
χ2=(40−1)(3.23)2(1.57)2\chi^{2}=\frac{(40-1) (3.23)^{2}}{(1.57)^{2}}

STEP 3

Calculate the numerator of the fraction, which is (n−1)s2(n-1) s^{2}.
Numerator=(40−1)(3.23)2Numerator = (40-1) (3.23)^{2}

STEP 4

Calculate the denominator of the fraction, which is σ2\sigma^{2}.
Denominator=(1.57)2Denominator = (1.57)^{2}

STEP 5

Calculate the value of the numerator.
Numerator=(40−1)(3.23)2=403.3821Numerator = (40-1) (3.23)^{2} = 403.3821

STEP 6

Calculate the value of the denominator.
Denominator=(1.57)2=2.4649Denominator = (1.57)^{2} = 2.4649

STEP 7

Now, divide the numerator by the denominator to find the value of χ2\chi^{2}.
χ2=NumeratorDenominator\chi^{2} = \frac{Numerator}{Denominator}

STEP 8

Substitute the values of the numerator and denominator into the equation.
χ2=403.38212.4649\chi^{2} = \frac{403.3821}{2.4649}

STEP 9

Calculate the value of χ2\chi^{2}.
χ2=403.38212.4649=163.692\chi^{2} = \frac{403.3821}{2.4649} = 163.692
The value of χ2\chi^{2} is 163.692, rounded to three decimal places.

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