Math / Data & Statistics

QuestionQuestion 13
In a recent poll, 580 people were asked if they liked dogs, and 68\% said they did. Find the Margin of Error for this poll, at the $99 \%$ confidence level. Give your answer to four decimal places if possible. $\square$

Studdy Solution

STEP 1

What is this asking? We need to figure out how much the poll results could be off by, given the number of people asked and the confidence level. Watch out! Don't mix up the percentage of people who like dogs with the margin of error!
We're not looking at how many people like dogs, but how accurate our measurement is.

STEP 2

1. Find the critical value.
2. Calculate the standard error.
3. Compute the margin of error.

STEP 3

For a 99% confidence level, we look up the z-score.
This tells us how many standard deviations away from the mean we need to go to capture 99% of the data.
It's like drawing a line in the sand and saying, "We're 99% sure the true percentage of dog-lovers is somewhere in here!"

STEP 4

For a 99% confidence level, our critical value, $\text{z}^*$ , is 2.576.
We can get this from a z-table or a calculator.

STEP 5

The standard error tells us how spread out our sample results are likely to be.
Think of it like the "average" amount our sample percentage might differ from the true percentage.

STEP 6

The formula for standard error is: $\text{SE} = \sqrt{\frac{p(1-p)}{n}}$ Where $p$ is the proportion from the sample (as a decimal) and $n$ is the number of people surveyed.

STEP 7

In our case, $p = 0.68$ (68% as a decimal) and $n = 580$ .
Let's plug those in: $\text{SE} = \sqrt{\frac{0.68 \cdot (1 - 0.68)}{580}}$ $\text{SE} = \sqrt{\frac{0.68 \cdot 0.32}{580}}$ $\text{SE} = \sqrt{\frac{0.2176}{580}}$ $\text{SE} \approx \sqrt{0.00037517}$ $\text{SE} \approx 0.01937$ So our standard error is approximately 0.01937.

STEP 8

Now, we combine our critical value and standard error to find the margin of error.
This tells us the range within which we're 99% confident the true percentage lies.

STEP 9

The formula for margin of error is: $\text{Margin of Error} = \text{z}^* \cdot \text{SE}$

STEP 10

Plugging in our values: $\text{Margin of Error} = 2.576 \cdot 0.01937$ $\text{Margin of Error} \approx 0.04991$

QuestionQuestion 13
In a recent poll, 580 people were asked if they liked dogs, and 68\% said they did. Find the Margin of Error for this poll, at the $99 \%$ confidence level. Give your answer to four decimal places if possible. $\square$

Studdy Solution

STEP 1

STEP 2

1. Find the critical value.
2. Calculate the standard error.
3. Compute the margin of error.

STEP 3

For a 99% confidence level, we look up the z-score.
This tells us how many standard deviations away from the mean we need to go to capture 99% of the data.
It's like drawing a line in the sand and saying, "We're 99% sure the true percentage of dog-lovers is somewhere in here!"

STEP 4

For a 99% confidence level, our critical value, $\text{z}^*$ , is 2.576.
We can get this from a z-table or a calculator.

STEP 5

The standard error tells us how spread out our sample results are likely to be.
Think of it like the "average" amount our sample percentage might differ from the true percentage.

STEP 6

The formula for standard error is: $\text{SE} = \sqrt{\frac{p(1-p)}{n}}$ Where $p$ is the proportion from the sample (as a decimal) and $n$ is the number of people surveyed.

STEP 7

STEP 8

Now, we combine our critical value and standard error to find the margin of error.
This tells us the range within which we're 99% confident the true percentage lies.

STEP 9

The formula for margin of error is: $\text{Margin of Error} = \text{z}^* \cdot \text{SE}$

STEP 10

Plugging in our values: $\text{Margin of Error} = 2.576 \cdot 0.01937$ $\text{Margin of Error} \approx 0.04991$

STEP 11

Rounding to four decimal places, we get 0.0499.

STEP 12

The margin of error is 0.0499.

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QuestionQuestion 13In a recent poll, 580 people were asked if they liked dogs, and 68\% said they did. Find the Margin of Error for this poll, at the 99%99 \%99% confidence level. Give your answer to four decimal places if possible. □\square□

Studdy Solution

STEP 1

STEP 2

1. Find the critical value.2. Calculate the standard error.3. Compute the margin of error.

STEP 3

For a 99% confidence level, we look up the z-score.This tells us how many standard deviations away from the mean we need to go to capture 99% of the data.It's like drawing a line in the sand and saying, "We're 99% sure the *true* percentage of dog-lovers is somewhere in here!"

STEP 4

For a 99% confidence level, our critical value, z∗\text{z}^*z∗, is **2.576**.We can get this from a z-table or a calculator.

STEP 5

The standard error tells us how spread out our sample results are likely to be.Think of it like the "average" amount our sample percentage might differ from the *true* percentage.

STEP 6

The formula for standard error is: SE=p(1−p)n \text{SE} = \sqrt{\frac{p(1-p)}{n}} SE=np(1−p)​​ Where ppp is the proportion from the sample (as a decimal) and nnn is the number of people surveyed.

STEP 7

STEP 8

Now, we combine our critical value and standard error to find the margin of error.This tells us the *range* within which we're 99% confident the true percentage lies.

STEP 9

The formula for margin of error is: Margin of Error=z∗⋅SE \text{Margin of Error} = \text{z}^* \cdot \text{SE} Margin of Error=z∗⋅SE

STEP 10

Plugging in our values: Margin of Error=2.576⋅0.01937 \text{Margin of Error} = 2.576 \cdot 0.01937 Margin of Error=2.576⋅0.01937 Margin of Error≈0.04991 \text{Margin of Error} \approx 0.04991 Margin of Error≈0.04991

STEP 11

Rounding to four decimal places, we get **0.0499**.

STEP 12

The margin of error is **0.0499**.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionQuestion 13
In a recent poll, 580 people were asked if they liked dogs, and 68\% said they did. Find the Margin of Error for this poll, at the $99 \%$ confidence level. Give your answer to four decimal places if possible. $\square$

1. Find the critical value.
2. Calculate the standard error.
3. Compute the margin of error.

For a 99% confidence level, we look up the z-score.
This tells us how many standard deviations away from the mean we need to go to capture 99% of the data.
It's like drawing a line in the sand and saying, "We're 99% sure the true percentage of dog-lovers is somewhere in here!"

For a 99% confidence level, our critical value, $\text{z}^*$ , is 2.576.
We can get this from a z-table or a calculator.

The standard error tells us how spread out our sample results are likely to be.
Think of it like the "average" amount our sample percentage might differ from the true percentage.

The formula for standard error is: $\text{SE} = \sqrt{\frac{p(1-p)}{n}}$ Where $p$ is the proportion from the sample (as a decimal) and $n$ is the number of people surveyed.

Now, we combine our critical value and standard error to find the margin of error.
This tells us the range within which we're 99% confident the true percentage lies.

The formula for margin of error is: $\text{Margin of Error} = \text{z}^* \cdot \text{SE}$

Plugging in our values: $\text{Margin of Error} = 2.576 \cdot 0.01937$ $\text{Margin of Error} \approx 0.04991$

Rounding to four decimal places, we get 0.0499.

The margin of error is 0.0499.