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A study randomly surveyed 1,000 teenagers to determine if they had a TV in their bedroom.
Surprisingly, 642 teens responded that they had a TV in their room. What is the margin of error for this study? Round to the nearest thousandth.

Sagot :

masifakrami

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.

semsee45

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.

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