Math / Data & Statistics

QuestionA poll in 2017 reported that 697 out of 1034 adults in a certain country believe that marijuana should be legalized. When this poll about the same subject was first conducted in 1969 , only $12 \%$ of the adults of the country supported legalization. Assume the conditions for using the CLT are met. Complete parts (a) through (d) below. a. Find and interpret a $99 \%$ confidence interval for the proportion of adults in the country in 2017 that believe marijuana should be legalized.
The 99\% confidence interval for the proportion of adults in the country in 2017 that believe marijuana should be legalized is ( $\square$ , (Round to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find a range of plausible values, with 99% confidence, for the true proportion of adults who support marijuana legalization in 2017, based on a poll. Watch out! Don't mix up the sample proportion with the true population proportion!
Also, remember to correctly interpret the meaning of the confidence interval.

STEP 2

1. Calculate the sample proportion.
2. Calculate the margin of error.
3. Construct the confidence interval.

STEP 3

Let's dive in!
We're given that 697 out of 1034 adults support legalization.
We can write this as a fraction to find the sample proportion, often denoted as $\hat{p}$ .

STEP 4

So, $\hat{p} = \frac{\text{Number of adults who support legalization}}{\text{Total number of adults polled}} = \frac{697}{1034}$ .

STEP 5

Calculating this gives us $\hat{p} \approx 0.674$ .
This means that approximately 67.4% of the polled adults supported legalization in 2017.

STEP 6

The margin of error tells us how much "wiggle room" we have around our sample proportion.
It's calculated using the formula $E = z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ , where $z^*$ is the critical value associated with our desired confidence level, $\hat{p}$ is our sample proportion, and $n$ is our sample size.

STEP 7

For a 99% confidence level, the critical value $z^*$ is 2.576.
We found our $\hat{p}$ to be approximately 0.674, and our sample size $n$ is 1034.

STEP 8

Plugging these values into our formula, we get $E = 2.576 \cdot \sqrt{\frac{0.674(1-0.674)}{1034}} \approx 2.576 \cdot \sqrt{\frac{0.674 \cdot 0.326}{1034}} \approx 2.576 \cdot \sqrt{\frac{0.2196}{1034}} \approx 2.576 \cdot \sqrt{0.000212} \approx 2.576 \cdot 0.0146 \approx 0.0376$ .

STEP 9

The confidence interval is calculated as $\hat{p} \pm E$ .
This means we take our sample proportion and add and subtract the margin of error to get a range of values.

STEP 10

Using our calculated values, the confidence interval is $0.674 \pm 0.0376$ .
This gives us a lower bound of $0.674 - 0.0376 = 0.6364$ and an upper bound of $0.674 + 0.0376 = 0.7116$ .

STEP 11

Rounding to three decimal places, our 99% confidence interval is $(0.636, 0.712)$ .

STEP 12

The 99% confidence interval for the proportion of adults in the country in 2017 that believe marijuana should be legalized is (0.636, 0.712).

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Studdy Solution

STEP 1

STEP 2

1. Calculate the sample proportion.
2. Calculate the margin of error.
3. Construct the confidence interval.

STEP 3

Let's dive in!
We're given that 697 out of 1034 adults support legalization.
We can write this as a fraction to find the sample proportion, often denoted as $\hat{p}$ .

STEP 4

So, $\hat{p} = \frac{\text{Number of adults who support legalization}}{\text{Total number of adults polled}} = \frac{697}{1034}$ .

STEP 5

Calculating this gives us $\hat{p} \approx 0.674$ .
This means that approximately 67.4% of the polled adults supported legalization in 2017.

STEP 6

STEP 7

For a 99% confidence level, the critical value $z^*$ is 2.576.
We found our $\hat{p}$ to be approximately 0.674, and our sample size $n$ is 1034.

STEP 8

Plugging these values into our formula, we get $E = 2.576 \cdot \sqrt{\frac{0.674(1-0.674)}{1034}} \approx 2.576 \cdot \sqrt{\frac{0.674 \cdot 0.326}{1034}} \approx 2.576 \cdot \sqrt{\frac{0.2196}{1034}} \approx 2.576 \cdot \sqrt{0.000212} \approx 2.576 \cdot 0.0146 \approx 0.0376$ .

STEP 9

The confidence interval is calculated as $\hat{p} \pm E$ .
This means we take our sample proportion and add and subtract the margin of error to get a range of values.

STEP 10

Using our calculated values, the confidence interval is $0.674 \pm 0.0376$ .
This gives us a lower bound of $0.674 - 0.0376 = 0.6364$ and an upper bound of $0.674 + 0.0376 = 0.7116$ .

STEP 11

Rounding to three decimal places, our 99% confidence interval is $(0.636, 0.712)$ .

STEP 12

The 99% confidence interval for the proportion of adults in the country in 2017 that believe marijuana should be legalized is (0.636, 0.712).

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Studdy Solution

STEP 1

STEP 2

1. Calculate the sample proportion.2. Calculate the margin of error.3. Construct the confidence interval.

STEP 3

Let's **dive in**!We're given that **697** out of **1034** adults support legalization.We can write this as a fraction to find the **sample proportion**, often denoted as p^\hat{p}p^​.

STEP 4

So, p^=Number of adults who support legalizationTotal number of adults polled=6971034\hat{p} = \frac{\text{Number of adults who support legalization}}{\text{Total number of adults polled}} = \frac{697}{1034}p^​=Total number of adults polledNumber of adults who support legalization​=1034697​.

STEP 5

Calculating this gives us p^≈0.674\hat{p} \approx 0.674p^​≈0.674.This means that approximately **67.4%** of the polled adults supported legalization in 2017.

STEP 6

STEP 7

For a **99% confidence level**, the critical value z∗z^*z∗ is **2.576**.We found our p^\hat{p}p^​ to be approximately **0.674**, and our sample size nnn is **1034**.

STEP 8

STEP 9

The **confidence interval** is calculated as p^±E\hat{p} \pm Ep^​±E.This means we take our **sample proportion** and add and subtract the **margin of error** to get a range of values.

STEP 10

Using our calculated values, the confidence interval is 0.674±0.03760.674 \pm 0.03760.674±0.0376.This gives us a lower bound of 0.674−0.0376=0.63640.674 - 0.0376 = 0.63640.674−0.0376=0.6364 and an upper bound of 0.674+0.0376=0.71160.674 + 0.0376 = 0.71160.674+0.0376=0.7116.

STEP 11

Rounding to three decimal places, our **99% confidence interval** is (0.636,0.712)(0.636, 0.712)(0.636,0.712).

STEP 12

The 99% confidence interval for the proportion of adults in the country in 2017 that believe marijuana should be legalized is (0.636, 0.712).

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

1. Calculate the sample proportion.
2. Calculate the margin of error.
3. Construct the confidence interval.

Let's dive in!
We're given that 697 out of 1034 adults support legalization.
We can write this as a fraction to find the sample proportion, often denoted as $\hat{p}$ .

So, $\hat{p} = \frac{\text{Number of adults who support legalization}}{\text{Total number of adults polled}} = \frac{697}{1034}$ .

Calculating this gives us $\hat{p} \approx 0.674$ .
This means that approximately 67.4% of the polled adults supported legalization in 2017.

For a 99% confidence level, the critical value $z^*$ is 2.576.
We found our $\hat{p}$ to be approximately 0.674, and our sample size $n$ is 1034.

The confidence interval is calculated as $\hat{p} \pm E$ .
This means we take our sample proportion and add and subtract the margin of error to get a range of values.

Using our calculated values, the confidence interval is $0.674 \pm 0.0376$ .
This gives us a lower bound of $0.674 - 0.0376 = 0.6364$ and an upper bound of $0.674 + 0.0376 = 0.7116$ .

Rounding to three decimal places, our 99% confidence interval is $(0.636, 0.712)$ .