Answer :
Answer:
The margin of error for this study is approximately 0.030 when rounded to the nearest thousandth.
Step-by-step explanation:
To calculate the margin of error for this study, we will use the formula for the margin of error for a proportion. The margin of error (ME) for a proportion is given by:
[tex]\[ME = Z \times \sqrt{\frac{p(1-p)}{n}}\][/tex]
where:
[tex]- \( Z \) is the Z-score corresponding to the desired confidence level.\\- \( p \) is the sample proportion.\\- \( n \) is the sample size.\\[/tex]
In this case, we need to determine the proportion of teenagers with a TV in their bedroom and then calculate the margin of error at a given confidence level (commonly 95%).
First, let's find the sample proportion p:
[tex]\[p = \frac{642}{1000} = 0.642\][/tex]
For a 95% confidence level, the Z-score Z is approximately 1.96.
Now, we can plug the values into the margin of error formula:
[tex]\[ME = 1.96 \times \sqrt{\frac{0.642 \times (1 - 0.642)}{1000}}\]\[ME = 1.96 \times \sqrt{\frac{0.642 \times 0.358}{1000}}\]\[ME = 1.96 \times \sqrt{\frac{0.229836}{1000}}\]\[ME = 1.96 \times \sqrt{0.000229836}\]\[ME = 1.96 \times 0.01516\]\[ME \approx 0.0297\][/tex]
Thus, the margin of error for this study is approximately 0.030 when rounded to the nearest thousandth.
Answer:
2.971%
Step-by-step explanation:
To determine the margin of error (MOE) for a proportion, we can use the formula for the margin of error of a proportion in a simple random sample:
[tex]MOE = z \times \sqrt{\dfrac{p(1-p)}{n}}[/tex]
where:
- z is the z-score.
- p is the sample proportion.
- n is the sample size.
In this case:
- p = 642/1000 = 0.642
- n = 1000
We will use a 95% confidence level, so the corresponding z-score is z = 1.96.
Substitute the values into the formula:
[tex]MOE = 1.96 \times \sqrt{\dfrac{0.642(1-0.642)}{1000}}\\\\\\\\MOE = 1.96 \times \sqrt{\dfrac{0.642(0.358)}{1000}}\\\\\\\\MOE = 1.96 \times \sqrt{\dfrac{0.229836}{1000}}\\\\\\\\MOE = 1.96 \times \sqrt{0.000229836}\\\\\\MOE = 1.96 \times0.015160343004\\\\\\MOE=0.029714272287...\\\\\\MOE=2.971\%\; \sf (nearest\;thousandth)[/tex]
Therefore, the margin of error for this study is 2.971%, rounded to the nearest thousandth.
