Question

which of the following z-values, standard deviations, and sample sizes produce a margin of error of 0.95?

$me = \frac{z \bullet s}{\sqrt{n}}$

• $z = 2.14; s = 4; n = 9$
• $z = 2.14; s = 4; n = 81$
• $z = 2.14; s = 16; n = 9$
• $z = 2.14; s = 16; n = 81$

Explanation:

Step1: Recall the margin of error formula

The formula for the margin of error (ME) is \( ME=\frac{z\cdot s}{\sqrt{n}} \). We need to calculate ME for each option and see which one equals 0.95.

Step2: Calculate ME for Option 1 (\( z = 2.14; s = 4; n = 9 \))

Substitute the values into the formula: \( ME=\frac{2.14\times4}{\sqrt{9}} \). First, calculate the denominator: \( \sqrt{9}=3 \). Then the numerator: \( 2.14\times4 = 8.56 \). Now divide: \( \frac{8.56}{3}\approx2.85 \). This is not 0.95.

Step3: Calculate ME for Option 2 (\( z = 2.14; s = 4; n = 81 \))

Substitute the values: \( ME=\frac{2.14\times4}{\sqrt{81}} \). Denominator: \( \sqrt{81}=9 \). Numerator: \( 2.14\times4 = 8.56 \). Divide: \( \frac{8.56}{9}\approx0.95 \) (rounded to two decimal places). Let's check the other options to be sure.

Step4: Calculate ME for Option 3 (\( z = 2.14; s = 16; n = 9 \))

Substitute: \( ME=\frac{2.14\times16}{\sqrt{9}} \). Denominator: 3. Numerator: \( 2.14\times16 = 34.24 \). Divide: \( \frac{34.24}{3}\approx11.41 \). Not 0.95.

Step5: Calculate ME for Option 4 (\( z = 2.14; s = 16; n = 81 \))

Substitute: \( ME=\frac{2.14\times16}{\sqrt{81}} \). Denominator: 9. Numerator: \( 2.14\times16 = 34.24 \). Divide: \( \frac{34.24}{9}\approx3.80 \). Not 0.95.

Answer:

\( z = 2.14; s = 4; n = 81 \) (the second option)