QuestionBelow, is the sample size, is the population proportion and is the sample proportion. Use the Central Limit Theorem and the calculator to find the probability. Round the answer to at least four decimal places.
Studdy Solution
STEP 1
What is this asking? What's the chance that our sample proportion is less than **0.22**, given a population proportion of **0.24**? Watch out! Don't forget to check if the Central Limit Theorem conditions are met!
STEP 2
1. Check Central Limit Theorem Applicability
2. Calculate the z-score
3. Find the Probability
STEP 3
Before we dive in, let's make sure we can actually *use* the Central Limit Theorem!
We need to check if and are both greater than or equal to **10**.
Since the problem doesn't give us , we'll assume it's large enough for the Central Limit Theorem to apply.
In a real-world scenario, you'd *always* want to make sure you have that value!
STEP 4
Alright, now for the z-score!
This tells us how far our sample proportion is from the population proportion, in terms of standard deviations.
The formula is:
STEP 5
We know and .
Since we are assuming is large, let's plug those values into our formula, still keeping as a variable:
STEP 6
Let's simplify the numerator: .
And the denominator: , so we have .
Now our equation looks like this:
STEP 7
We can rewrite the fraction in the denominator as a product with :
STEP 8
Now, we need to find , which is the same as finding .
Since we're assuming a large , this probability will be very close to the population proportion .
We're asked to round to four decimal places, so let's use a "very large" like .
STEP 9
With , we have .
STEP 10
Using a calculator or a z-table, we find that is a very small number, approximately **0.0000**.
STEP 11
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