Question
Studdy Solution
STEP 1
What is this asking?
We want to find the range where we're 90% sure the *true* average age of all U.S. college students falls, based on a sample of 150 students.
Watch out!
Don't mix up the sample mean with the population mean!
We're using the sample to *estimate* the population mean.
Also, make sure to use the correct z-score for a 90% confidence interval.
STEP 2
1. Find the Critical Value
2. Calculate the Margin of Error
3. Compute the Confidence Interval
STEP 3
We're dealing with a 90% confidence interval, which means there's 10% *outside* the interval.
Since the normal distribution is symmetric, this 10% is split evenly into 5% in each tail.
So, we're looking for the z-score that leaves 5% in the right tail, or 95% to the left.
STEP 4
This special z-score is called the **critical value**, and for a 90% confidence level, it's approximately .
You can find this using a z-table or a calculator!
STEP 5
The **margin of error** tells us how much "wiggle room" we have around our sample mean.
The formula for the margin of error is:
where is our **critical value**, is the **population standard deviation**, and is our **sample size**.
STEP 6
Let's plug in our values!
We have , , and .
STEP 7
Time to crunch the numbers! So, our margin of error is approximately **0.594** years.
STEP 8
The confidence interval is calculated by taking our **sample mean** and adding/subtracting the **margin of error**.
Our sample mean is .
STEP 9
The lower bound of our interval is: The upper bound of our interval is:
STEP 10
Rounding to two decimal places, our 90% confidence interval is .
STEP 11
We are 90% confident that the true mean age of U.S. college students is between 22.18 and 23.36 years old.
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