Math  /  Data & Statistics

QuestionAn NHANES report gives data for 936 men aged 20-29 years. The BMI of these 936 men was xˉ=27.2\bar{x}=27.2. On the basis of this sample, we want to estimate the BMI μ\mu in the population of all 23.2 million American men in this age group. To match the "simple conditions," we will treat the NHANES sample as an SRS from a Normal population with standard deviation σ=11.6\sigma=11.6. Healdh Statistics 3(2016), at hutps/lwww.cdegov/nchs/data/ series/sr_03/sr03_039.pdf.
What are the margins of errors for 90%,95%90 \%, 95 \%, and 99%99 \% confidence? Give your answers to four decimal places.

Studdy Solution

STEP 1

What is this asking? We want to find the margin of error for the average BMI of young American men, given data from a sample, at different confidence levels. Watch out! Don't mix up standard deviation (σ\sigma) and standard error (SESE).
Also, remember each confidence level has a specific z-score.

STEP 2

1. Calculate the Standard Error
2. Calculate the Margin of Error for 90% Confidence
3. Calculate the Margin of Error for 95% Confidence
4. Calculate the Margin of Error for 99% Confidence

STEP 3

Alright, let's **kick things off** by finding the **standard error**!
This tells us how much our sample mean is likely to vary from the true population mean.
The formula for standard error is SE=σnSE = \frac{\sigma}{\sqrt{n}}, where σ\sigma is the **population standard deviation** and nn is the **sample size**.

STEP 4

We're given σ=11.6\sigma = 11.6 and n=936n = 936.
Let's **plug those values** into our formula: SE=11.6936SE = \frac{11.6}{\sqrt{936}}.

STEP 5

**Crunching the numbers**, we get SE0.3802SE \approx 0.3802.
So, our standard error is approximately **0.3802**.
This means our sample mean is likely within 0.3802 of the true population mean.

STEP 6

Now, let's find the **margin of error** for a **90% confidence level**.
The margin of error tells us how much our estimate is likely to vary from the true value.
The formula is ME=zSEME = z^* \cdot SE, where zz^* is the **critical value** for our chosen confidence level.

STEP 7

For a 90% confidence level, z=1.645z^* = 1.645.
We already calculated our SESE as approximately 0.38020.3802. **Plugging these values** into our formula, we get ME=1.6450.3802ME = 1.645 \cdot 0.3802.

STEP 8

Calculating this gives us ME0.6255ME \approx 0.6255.
So, our **margin of error** for 90% confidence is approximately **0.6255**.

STEP 9

Let's **repeat the process** for a **95% confidence level**!
The only thing that changes is our zz^* value.

STEP 10

For 95% confidence, z=1.96z^* = 1.96.
Using our SESE of approximately 0.38020.3802, our formula becomes ME=1.960.3802ME = 1.96 \cdot 0.3802.

STEP 11

This gives us ME0.7452ME \approx 0.7452.
Our **margin of error** for 95% confidence is approximately **0.7452**.

STEP 12

Finally, let's tackle the **99% confidence level**.

STEP 13

For 99% confidence, z=2.576z^* = 2.576.
Our margin of error formula becomes ME=2.5760.3802ME = 2.576 \cdot 0.3802.

STEP 14

Calculating this, we get ME0.9794ME \approx 0.9794.
Our **margin of error** for 99% confidence is approximately **0.9794**.

STEP 15

The margins of error are approximately **0.6255** for 90% confidence, **0.7452** for 95% confidence, and **0.9794** for 99% confidence.

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