Math  /  Data & Statistics

QuestionHomework Part 1 of 4 Points: 0 of 1
According to a poll, 718 out of 1072 randomly selected adults living in a certain country felt the laws covering the sale of firearms should be more strict. a. What is the value of p^\hat{p}, the sample proportion who favor stricter gun laws? b. Check the conditions to determine whether the CLT can be used to find a confidence interval. c. Find a 95%95 \% confidence interval for the population proportion who favor stricter gun laws. d. Based on your confidence interval, do a majority of adults in the country favor stricter gun laws? a. The value of p^\hat{p}, the sample proportion who favor stricter gun laws, is \square (Round to two decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? We're looking at how many people want stricter gun laws, figuring out how sure we are about our results, and if most people agree. Watch out! Don't mix up the sample and the whole population!
Also, make sure we're allowed to use our confidence interval tricks.

STEP 2

1. Calculate Sample Proportion
2. Check CLT Conditions
3. Calculate Confidence Interval
4. Interpret the Confidence Interval

STEP 3

Alright, let's **dive in**!
We've got **718** people out of **1072** who want stricter laws.
To get the **sample proportion**, we just divide the number who want stricter laws by the total number of people surveyed.
It's like figuring out what fraction of a pizza you ate!

STEP 4

So, our calculation is: p^=71810720.67 \hat{p} = \frac{\textbf{718}}{\textbf{1072}} \approx \textbf{0.67} This means about **67%** of the *sampled* people want stricter gun laws.

STEP 5

Before we get too excited and start making big claims, we need to check if we're allowed to use the **Central Limit Theorem (CLT)**.
It's like making sure you have enough ingredients before you start baking a cake!

STEP 6

The CLT has a few rules.
First, our sample needs to be random, which it is!
Second, we need to check if np^ n \cdot \hat{p} and n(1p^) n \cdot (1 - \hat{p}) are both greater than or equal to **10**.
This makes sure our sample is big enough to play nice with the CLT.

STEP 7

Let's do the math! np^=10720.67718 n \cdot \hat{p} = 1072 \cdot 0.67 \approx 718 n(1p^)=1072(10.67)=10720.33354 n \cdot (1 - \hat{p}) = 1072 \cdot (1 - 0.67) = 1072 \cdot 0.33 \approx 354 Since both **718** and **354** are way bigger than **10**, we're good to go!
The CLT says "Party on!"

STEP 8

Now for the main event: the **95% confidence interval**!
This tells us a range where we're pretty sure the *true* proportion of people who want stricter laws lies.

STEP 9

The formula for a confidence interval is: p^±zp^(1p^)n \hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} Where p^ \hat{p} is our **sample proportion**, z z^* is a special number based on our **confidence level** (for 95%, it's about **1.96**), and n n is our **sample size**.

STEP 10

Let's plug in the numbers: 0.67±1.960.67(10.67)1072 0.67 \pm 1.96 \cdot \sqrt{\frac{0.67 \cdot (1 - 0.67)}{1072}} 0.67±1.960.670.331072 0.67 \pm 1.96 \cdot \sqrt{\frac{0.67 \cdot 0.33}{1072}} 0.67±1.960.22111072 0.67 \pm 1.96 \cdot \sqrt{\frac{0.2211}{1072}} 0.67±1.960.000206 0.67 \pm 1.96 \cdot \sqrt{0.000206} 0.67±1.960.014 0.67 \pm 1.96 \cdot 0.014 0.67±0.027 0.67 \pm 0.027

STEP 11

So, our **95% confidence interval** is from 0.670.027=0.6430.67 - 0.027 = \textbf{0.643} to 0.67+0.027=0.6970.67 + 0.027 = \textbf{0.697}.

STEP 12

We're **95% confident** that the *true* proportion of adults who want stricter gun laws is somewhere between **64.3%** and **69.7%**.

STEP 13

Since our entire confidence interval is *above* **50%**, we can say yes, it's likely that a majority of adults in the country favor stricter gun laws!

STEP 14

a. p^0.67\hat{p} \approx 0.67 b. The CLT conditions are met. c. The 95% confidence interval is approximately (0.643, 0.697). d. Yes, a majority likely favor stricter gun laws.

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