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Math  /  Algebra

QuestionEvaluate the following expression at the values xˉ=107.2,μ=103,σ=0.91\bar{x}=107.2, \mu=103, \sigma=0.91, and n=161n=161 xˉ−μσn\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}
Enter your answer as a decimal rounded to four decimal places. □\square

Studdy Solution

STEP 1

1. The expression to be evaluated is a form of the z-score or standard score.
2. xˉ\bar{x} represents the sample mean.
3. μ\mu represents the population mean.
4. σ\sigma represents the population standard deviation.
5. nn represents the sample size.
6. All values provided are real numbers, and the result should be a real number rounded to four decimal places.

STEP 2

1. Substitute the given values into the expression.
2. Simplify the denominator.
3. Perform the division.
4. Round the result to four decimal places.

STEP 3

Substitute the given values into the expression: xˉ=107.2,μ=103,σ=0.91,n=161 \bar{x} = 107.2, \mu = 103, \sigma = 0.91, n = 161 xˉ−μσn=107.2−1030.91161 \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} = \frac{107.2 - 103}{\frac{0.91}{\sqrt{161}}}

STEP 4

Simplify the denominator by calculating n\sqrt{n}: 161≈12.6886 \sqrt{161} \approx 12.6886

STEP 5

Continue simplifying the denominator by dividing σ\sigma by n\sqrt{n}: 0.9112.6886≈0.0717 \frac{0.91}{12.6886} \approx 0.0717

STEP 6

Perform the subtraction in the numerator: 107.2−103=4.2 107.2 - 103 = 4.2

STEP 7

Divide the numerator by the simplified denominator: 4.20.0717≈58.5706 \frac{4.2}{0.0717} \approx 58.5706

STEP 8

Round the result to four decimal places: 58.5706≈58.5706 58.5706 \approx 58.5706
Solution: 58.5706 \boxed{58.5706}

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