Math / Data & Statistics

Question1. Consider a normal population distribution with the value of $\sigma$ known. a. What is the confidence level for the interval $\bar{x} \pm 2.81 \sigma / \sqrt{n}$ ? b. What is the confidence level for the interval $\bar{x} \pm 1.44 \sigma / \sqrt{n}$ ? c. What value of $z_{\alpha/2}$ in the CI formula (7.5) results in a confidence level of 99.7%? d. Answer the question posed in part (c) for a confidence level of 75%.

Studdy Solution

STEP 1

What is this asking? We're figuring out the confidence levels and z-scores for different confidence intervals when we already know the population standard deviation! Watch out! Don't mix up the z-score and the confidence level!
They're related, but not the same thing.

STEP 2

1. Confidence Level for 2.81
2. Confidence Level for 1.44
3. Z-score for 99.7%
4. Z-score for 75%

STEP 3

Alright, so we're given this interval: $\bar{x} \pm 2.81 \sigma / \sqrt{n}$ .
This looks a lot like a confidence interval formula!
The critical part here is the 2.81, which is our $z_{\alpha/2}$ value.

STEP 4

We need to find $\alpha/2$ .
Since $z_{\alpha/2} = \textbf{2.81}$ , we can look up this value in a z-table (or use a calculator) to find the area to the right of 2.81.
This area represents $\alpha/2$ .
We find that $\alpha/2 \approx \textbf{0.0025}$ .

STEP 5

Now, we need $\alpha$ , so we multiply $\alpha/2$ by 2: $\alpha = 2 \cdot \textbf{0.0025} = \textbf{0.005}$ .

STEP 6

Finally, the confidence level is $1 - \alpha$ .
So, $1 - \textbf{0.005} = \textbf{0.995}$ .
Converting to a percentage, we get a 99.5% confidence level!

STEP 7

This time, our interval is $\bar{x} \pm 1.44 \sigma / \sqrt{n}$ , so $z_{\alpha/2} = \textbf{1.44}$ .

STEP 8

Looking up $\textbf{1.44}$ in our z-table (or using a calculator), we find the area to the right, which is $\alpha/2 \approx \textbf{0.0749}$ .

STEP 9

Multiply by 2 to get $\alpha$ : $\alpha = 2 \cdot \textbf{0.0749} = \textbf{0.1498}$ .

STEP 10

The confidence level is $1 - \alpha = 1 - \textbf{0.1498} = \textbf{0.8502}$ .
That's an 85.02% confidence level!

STEP 11

Now, we're given the confidence level and need to find $z_{\alpha/2}$ .
Our confidence level is 99.7%, or $\textbf{0.997}$ .

STEP 12

Since the confidence level is $1 - \alpha$ , we have $\alpha = 1 - \textbf{0.997} = \textbf{0.003}$ .

STEP 13

We need $\alpha/2$ , so divide by 2: $\alpha/2 = \textbf{0.003} / 2 = \textbf{0.0015}$ .

STEP 14

Now, we look up the z-score that corresponds to an area of $\textbf{0.0015}$ to its right in the z-table.
This gives us $z_{\alpha/2} \approx \textbf{2.97}$ .

STEP 15

Our confidence level is 75%, or $\textbf{0.75}$ .

STEP 16

So, $\alpha = 1 - \textbf{0.75} = \textbf{0.25}$ .

STEP 17

Then, $\alpha/2 = \textbf{0.25} / 2 = \textbf{0.125}$ .

STEP 18

Looking up the z-score corresponding to an area of $\textbf{0.125}$ to the right, we find $z_{\alpha/2} \approx \textbf{1.15}$ .

STEP 19

a. 99.5% b. 85.02% c. $z_{\alpha/2} \approx \textbf{2.97}$ d. $z_{\alpha/2} \approx \textbf{1.15}$

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Studdy Solution

STEP 1

What is this asking? We're figuring out the confidence levels and z-scores for different confidence intervals when we already know the population standard deviation! Watch out! Don't mix up the z-score and the confidence level!They're related, but not the same thing.

STEP 2

1. Confidence Level for 2.812. Confidence Level for 1.443. Z-score for 99.7%4. Z-score for 75%

STEP 3

Alright, so we're given this interval: xˉ±2.81σ/n\bar{x} \pm 2.81 \sigma / \sqrt{n}xˉ±2.81σ/n​.This looks a lot like a confidence interval formula!The critical part here is the **2.81**, which is our zα/2z_{\alpha/2}zα/2​ value.

STEP 4

STEP 5

Now, we need α\alphaα, so we multiply α/2\alpha/2α/2 by 2: α=2⋅0.0025=0.005\alpha = 2 \cdot \textbf{0.0025} = \textbf{0.005}α=2⋅0.0025=0.005.

STEP 6

Finally, the confidence level is 1−α1 - \alpha1−α.So, 1−0.005=0.9951 - \textbf{0.005} = \textbf{0.995}1−0.005=0.995.Converting to a percentage, we get a **99.5%** confidence level!

STEP 7

This time, our interval is xˉ±1.44σ/n\bar{x} \pm 1.44 \sigma / \sqrt{n}xˉ±1.44σ/n​, so zα/2=1.44z_{\alpha/2} = \textbf{1.44}zα/2​=1.44.

STEP 8

Looking up 1.44\textbf{1.44}1.44 in our z-table (or using a calculator), we find the area to the right, which is α/2≈0.0749\alpha/2 \approx \textbf{0.0749}α/2≈0.0749.

STEP 9

Multiply by 2 to get α\alphaα: α=2⋅0.0749=0.1498\alpha = 2 \cdot \textbf{0.0749} = \textbf{0.1498}α=2⋅0.0749=0.1498.

STEP 10

The confidence level is 1−α=1−0.1498=0.85021 - \alpha = 1 - \textbf{0.1498} = \textbf{0.8502}1−α=1−0.1498=0.8502.That's an **85.02%** confidence level!

STEP 11

Now, we're given the confidence level and need to find zα/2z_{\alpha/2}zα/2​.Our confidence level is **99.7%**, or 0.997\textbf{0.997}0.997.

STEP 12

Since the confidence level is 1−α1 - \alpha1−α, we have α=1−0.997=0.003\alpha = 1 - \textbf{0.997} = \textbf{0.003}α=1−0.997=0.003.

STEP 13

We need α/2\alpha/2α/2, so divide by 2: α/2=0.003/2=0.0015\alpha/2 = \textbf{0.003} / 2 = \textbf{0.0015}α/2=0.003/2=0.0015.

STEP 14

Now, we look up the z-score that corresponds to an area of 0.0015\textbf{0.0015}0.0015 to its *right* in the z-table.This gives us zα/2≈2.97z_{\alpha/2} \approx \textbf{2.97}zα/2​≈2.97.

STEP 15

Our confidence level is **75%**, or 0.75\textbf{0.75}0.75.

STEP 16

So, α=1−0.75=0.25\alpha = 1 - \textbf{0.75} = \textbf{0.25}α=1−0.75=0.25.

STEP 17

Then, α/2=0.25/2=0.125\alpha/2 = \textbf{0.25} / 2 = \textbf{0.125}α/2=0.25/2=0.125.

STEP 18

Looking up the z-score corresponding to an area of 0.125\textbf{0.125}0.125 to the right, we find zα/2≈1.15z_{\alpha/2} \approx \textbf{1.15}zα/2​≈1.15.

STEP 19

a. **99.5%** b. **85.02%** c. zα/2≈2.97z_{\alpha/2} \approx \textbf{2.97}zα/2​≈2.97 d. zα/2≈1.15z_{\alpha/2} \approx \textbf{1.15}zα/2​≈1.15

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What is this asking? We're figuring out the confidence levels and z-scores for different confidence intervals when we already know the population standard deviation! Watch out! Don't mix up the z-score and the confidence level!
They're related, but not the same thing.

1. Confidence Level for 2.81
2. Confidence Level for 1.44
3. Z-score for 99.7%
4. Z-score for 75%

Alright, so we're given this interval: $\bar{x} \pm 2.81 \sigma / \sqrt{n}$ .
This looks a lot like a confidence interval formula!
The critical part here is the 2.81, which is our $z_{\alpha/2}$ value.

Now, we need $\alpha$ , so we multiply $\alpha/2$ by 2: $\alpha = 2 \cdot \textbf{0.0025} = \textbf{0.005}$ .

Finally, the confidence level is $1 - \alpha$ .
So, $1 - \textbf{0.005} = \textbf{0.995}$ .
Converting to a percentage, we get a 99.5% confidence level!

This time, our interval is $\bar{x} \pm 1.44 \sigma / \sqrt{n}$ , so $z_{\alpha/2} = \textbf{1.44}$ .

Looking up $\textbf{1.44}$ in our z-table (or using a calculator), we find the area to the right, which is $\alpha/2 \approx \textbf{0.0749}$ .

Multiply by 2 to get $\alpha$ : $\alpha = 2 \cdot \textbf{0.0749} = \textbf{0.1498}$ .

The confidence level is $1 - \alpha = 1 - \textbf{0.1498} = \textbf{0.8502}$ .
That's an 85.02% confidence level!

Now, we're given the confidence level and need to find $z_{\alpha/2}$ .
Our confidence level is 99.7%, or $\textbf{0.997}$ .

Since the confidence level is $1 - \alpha$ , we have $\alpha = 1 - \textbf{0.997} = \textbf{0.003}$ .

We need $\alpha/2$ , so divide by 2: $\alpha/2 = \textbf{0.003} / 2 = \textbf{0.0015}$ .

Now, we look up the z-score that corresponds to an area of $\textbf{0.0015}$ to its right in the z-table.
This gives us $z_{\alpha/2} \approx \textbf{2.97}$ .

Our confidence level is 75%, or $\textbf{0.75}$ .

So, $\alpha = 1 - \textbf{0.75} = \textbf{0.25}$ .

Then, $\alpha/2 = \textbf{0.25} / 2 = \textbf{0.125}$ .

Looking up the z-score corresponding to an area of $\textbf{0.125}$ to the right, we find $z_{\alpha/2} \approx \textbf{1.15}$ .

a. 99.5% b. 85.02% c. $z_{\alpha/2} \approx \textbf{2.97}$ d. $z_{\alpha/2} \approx \textbf{1.15}$