Math / Data & Statistics

QuestionAccording to a poll, 674 out of 1061 randomly selected smokers polled believed they are discriminated against in public life or in employment because of their smoking a. What percentage of the smokers polled believed they are discriminated against because of their smoking? b. Check the conditions to determine whether the CLT can be used to find a confidence interval. c. Find a $95 \%$ confidence interval for the population proportion of smokers who believe they are discriminated against because of their smoking. d. Can this confidence interval be used to conclude that the majority of smokers believe they are discriminated against because of their smoking? Why or why non? a. The percentage of those taking the poll believed they are discriminated against because of their smoking is (Round to one decimal place as needed.) $\square$ \%. b. Check the conditions to determine whether you can apply the CLT to find a confidence interval.
The Random and Independent condition $\square$ reasonably be assumed to hold. The Large Sample condition $\square$ The Big Population condition $\square$ c. The $95 \%$ confidence interval is $\square$ . ). (Round to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We're looking at how many smokers feel discriminated against, turning that into a percentage, and then seeing how confident we are about that percentage across all smokers. Watch out! Don't mix up the number of people in the poll with the percentage of smokers who feel discriminated against.
Also, remember that a confidence interval isn't about individual smokers, but about how sure we are about our percentage estimate!

STEP 2

1. Calculate the Sample Proportion
2. Check CLT Conditions
3. Calculate the Margin of Error
4. Calculate the Confidence Interval
5. Interpret the Confidence Interval

STEP 3

Let's dive right in and find the percentage of smokers in the poll who feel discriminated against.
We've got 674 smokers out of 1061 total who feel this way.
To get the percentage, we'll divide the number of smokers who feel discriminated against by the total number of smokers polled, and then multiply by 100!

STEP 4

So, the calculation looks like this: $\frac{674}{1061} \cdot 100 \approx 63.52$ That means 63.5% of the smokers polled felt discriminated against.
Remember that 63.5%!
We'll need it later.

STEP 5

Before we get too excited, we need to check if we can use the Central Limit Theorem (CLT).
The CLT helps us use information from our sample to make guesses about all smokers.
It has a few conditions.

STEP 6

The problem says the smokers were randomly selected, so we're good here!
This means each person's answer doesn't affect anyone else's.

STEP 7

We need to check if our sample is large enough.
We do this by making sure both $n \cdot \hat{p}$ and $n \cdot (1 - \hat{p})$ are greater than or equal to 10.
Here, $n$ is our sample size (1061) and $\hat{p}$ is our sample proportion (0.6352, remember we found this earlier by dividing 674 by 1061).

STEP 8

Let's calculate: $n \cdot \hat{p} = 1061 \cdot 0.6352 \approx 674$ $n \cdot (1 - \hat{p}) = 1061 \cdot (1 - 0.6352) \approx 387$ Both are way bigger than 10, so we're good to go!

STEP 9

We can assume there are way more smokers than the 1061 we polled, so this condition is met too!

STEP 10

The margin of error tells us how much our sample percentage might differ from the real percentage for all smokers.
For a 95% confidence interval, we use a z-score of 1.96.
The formula for margin of error is: $\text{Margin of Error} = z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

STEP 11

Plugging in our values, we get: $1.96 \cdot \sqrt{\frac{0.6352 \cdot (1 - 0.6352)}{1061}} \approx 1.96 \cdot \sqrt{\frac{0.2322}{1061}} \approx 1.96 \cdot 0.0149 \approx 0.029$ So, our margin of error is about 0.029.

STEP 12

Now, we'll use our sample proportion and margin of error to find the confidence interval.
We simply add and subtract the margin of error from the sample proportion.

STEP 13

$0.6352 - 0.029 = 0.606$ $0.6352 + 0.029 = 0.664$ This gives us a 95% confidence interval of (0.606, 0.664).

STEP 14

This means we're 95% confident that the real percentage of all smokers who feel discriminated against is somewhere between 60.6% and 66.4%.

STEP 15

Since this entire interval is above 50%, we can say yes, the majority of smokers likely feel discriminated against!

STEP 16

a. 63.5% b. All conditions are met. c. (0.606, 0.664) d. Yes, because the entire confidence interval is greater than 50%.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Calculate the Sample Proportion
2. Check CLT Conditions
3. Calculate the Margin of Error
4. Calculate the Confidence Interval
5. Interpret the Confidence Interval

STEP 3

STEP 4

So, the calculation looks like this: $\frac{674}{1061} \cdot 100 \approx 63.52$ That means 63.5% of the smokers polled felt discriminated against.
Remember that 63.5%!
We'll need it later.

STEP 5

Before we get too excited, we need to check if we can use the Central Limit Theorem (CLT).
The CLT helps us use information from our sample to make guesses about all smokers.
It has a few conditions.

STEP 6

The problem says the smokers were randomly selected, so we're good here!
This means each person's answer doesn't affect anyone else's.

STEP 7

We need to check if our sample is large enough.
We do this by making sure both $n \cdot \hat{p}$ and $n \cdot (1 - \hat{p})$ are greater than or equal to 10.
Here, $n$ is our sample size (1061) and $\hat{p}$ is our sample proportion (0.6352, remember we found this earlier by dividing 674 by 1061).

STEP 8

Let's calculate: $n \cdot \hat{p} = 1061 \cdot 0.6352 \approx 674$ $n \cdot (1 - \hat{p}) = 1061 \cdot (1 - 0.6352) \approx 387$ Both are way bigger than 10, so we're good to go!

STEP 9

We can assume there are way more smokers than the 1061 we polled, so this condition is met too!

STEP 10

The margin of error tells us how much our sample percentage might differ from the real percentage for all smokers.
For a 95% confidence interval, we use a z-score of 1.96.
The formula for margin of error is: $\text{Margin of Error} = z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

STEP 11

Plugging in our values, we get: $1.96 \cdot \sqrt{\frac{0.6352 \cdot (1 - 0.6352)}{1061}} \approx 1.96 \cdot \sqrt{\frac{0.2322}{1061}} \approx 1.96 \cdot 0.0149 \approx 0.029$ So, our margin of error is about 0.029.

STEP 12

Now, we'll use our sample proportion and margin of error to find the confidence interval.
We simply add and subtract the margin of error from the sample proportion.

STEP 13

$0.6352 - 0.029 = 0.606$ $0.6352 + 0.029 = 0.664$ This gives us a 95% confidence interval of (0.606, 0.664).

STEP 14

This means we're 95% confident that the real percentage of all smokers who feel discriminated against is somewhere between 60.6% and 66.4%.

STEP 15

Since this entire interval is above 50%, we can say yes, the majority of smokers likely feel discriminated against!

STEP 16

a. 63.5% b. All conditions are met. c. (0.606, 0.664) d. Yes, because the entire confidence interval is greater than 50%.

Get Studdy

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.

Studdy Solution

STEP 1

STEP 2

1. Calculate the Sample Proportion2. Check CLT Conditions3. Calculate the Margin of Error4. Calculate the Confidence Interval5. Interpret the Confidence Interval

STEP 3

STEP 4

So, the calculation looks like this: 6741061⋅100≈63.52 \frac{674}{1061} \cdot 100 \approx 63.52 1061674​⋅100≈63.52 That means **63.5%** of the smokers polled felt discriminated against.Remember that **63.5%**!We'll need it later.

STEP 5

Before we get too excited, we need to check if we can use the Central Limit Theorem (CLT).The CLT helps us use information from our sample to make guesses about *all* smokers.It has a few conditions.

STEP 6

The problem says the smokers were randomly selected, so we're good here!This means each person's answer doesn't affect anyone else's.

STEP 7

STEP 8

STEP 9

We can assume there are way more smokers than the **1061** we polled, so this condition is met too!

STEP 10

STEP 11

STEP 12

Now, we'll use our sample proportion and margin of error to find the confidence interval.We simply add and subtract the margin of error from the sample proportion.

STEP 13

0.6352−0.029=0.606 0.6352 - 0.029 = 0.606 0.6352−0.029=0.606 0.6352+0.029=0.664 0.6352 + 0.029 = 0.664 0.6352+0.029=0.664 This gives us a 95% confidence interval of (**0.606**, **0.664**).

STEP 14

This means we're 95% confident that the *real* percentage of all smokers who feel discriminated against is somewhere between **60.6%** and **66.4%**.

STEP 15

Since this entire interval is above 50%, we can say yes, the majority of smokers likely feel discriminated against!

STEP 16

a. **63.5%** b. All conditions are met. c. (**0.606**, **0.664**) d. Yes, because the entire confidence interval is greater than 50%.

Get Studdy

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.

1. Calculate the Sample Proportion
2. Check CLT Conditions
3. Calculate the Margin of Error
4. Calculate the Confidence Interval
5. Interpret the Confidence Interval

So, the calculation looks like this: $\frac{674}{1061} \cdot 100 \approx 63.52$ That means 63.5% of the smokers polled felt discriminated against.
Remember that 63.5%!
We'll need it later.

Before we get too excited, we need to check if we can use the Central Limit Theorem (CLT).
The CLT helps us use information from our sample to make guesses about all smokers.
It has a few conditions.

The problem says the smokers were randomly selected, so we're good here!
This means each person's answer doesn't affect anyone else's.

We can assume there are way more smokers than the 1061 we polled, so this condition is met too!

Now, we'll use our sample proportion and margin of error to find the confidence interval.
We simply add and subtract the margin of error from the sample proportion.

$0.6352 - 0.029 = 0.606$ $0.6352 + 0.029 = 0.664$ This gives us a 95% confidence interval of (0.606, 0.664).

This means we're 95% confident that the real percentage of all smokers who feel discriminated against is somewhere between 60.6% and 66.4%.

a. 63.5% b. All conditions are met. c. (0.606, 0.664) d. Yes, because the entire confidence interval is greater than 50%.