Answered

C Math 3

A recent poll of 500 employees from a company of 1,300 employees was conducted to see how many of them believe the minimum wage should be raised. Of those polled, 435 feel that the minimum wage should be raised. With a desired confidence level of [tex]$95 \%$[/tex], and a corresponding [tex]$z^\ \textless \ em\ \textgreater \ $[/tex]-score of 1.96, what is the margin of error for this sample survey?

Complete the statements to find the margin of error:

1. The sample size in this problem is [tex]\square[/tex] employees.
2. The population proportion is estimated as [tex]\square[/tex].
3. When the margin of error is calculated using the formula [tex]E=z^\ \textless \ /em\ \textgreater \ \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/tex], to the nearest tenth of a percent, the result is [tex]\square \checkmark \%[/tex].



Answer :

GinnyAnswer
Certainly! Let's solve this step-by-step:

1. Identify the sample size:

The sample size is the number of employees polled. Here, the sample size is 500.

[tex]\[ \boxed{500} \][/tex]

2. Calculate the sample proportion ([tex]\(\hat{p}\)[/tex]):

The sample proportion is the number of employees who feel that the minimum wage should be raised divided by the total number of employees polled. Here, 435 out of 500 employees support the raise.

[tex]\[ \hat{p} = \frac{435}{500} = 0.87 \][/tex]

So, the population proportion is estimated as:

[tex]\[ \boxed{0.87} \][/tex]

3. Calculate the margin of error (E):

We use the formula for the margin of error for a proportion:

[tex]\[ E = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \][/tex]

Here:
- [tex]\( z^* \)[/tex] is the z-score corresponding to the 95% confidence level, which is 1.96.
- [tex]\( \hat{p} \)[/tex] is 0.87.
- [tex]\( n \)[/tex] is the sample size, which is 500.

Plugging in these values:

[tex]\[ E = 1.96 \cdot \sqrt{\frac{0.87 \cdot (1 - 0.87)}{500}} \][/tex]

Simplifying inside the square root first:

[tex]\[ E = 1.96 \cdot \sqrt{\frac{0.87 \cdot 0.13}{500}} \][/tex]

Further simplifying:

[tex]\[ E = 1.96 \cdot \sqrt{\frac{0.1131}{500}} \][/tex]

[tex]\[ E = 1.96 \cdot \sqrt{0.0002262} \][/tex]

[tex]\[ E = 1.96 \cdot 0.015034792 \][/tex]

Multiplying by 1.96:

[tex]\[ E = 0.029478295744496494 \][/tex]

Converting the margin of error to a percentage:

[tex]\[ 0.029478295744496494 \times 100 = 2.9478295744496494\% \][/tex]

Rounding to the nearest tenth of a percent:

[tex]\[ 2.9\% \][/tex]

So, the margin of error is:

[tex]\[ \boxed{2.9\%} \][/tex]

Other Questions