Question Find the margin of error for $c=0.95$ , $s=4$ , and $n=8$ .

Studdy Solution

STEP 1

Assumptions
1. The confidence level ( $c$ ) is 0.95.
2. The sample standard deviation ( $s$ ) is 4.
3. The sample size ( $n$ ) is 8.
4. The margin of error is calculated using the formula for a t-distribution, as the population standard deviation is unknown and the sample size is small ( $n < 30$ ).
5. The margin of error formula for a t-distribution is given by: $E = t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}}$ where $t_{\frac{\alpha}{2}}$ is the t-score that corresponds to the confidence level and degrees of freedom ( $df = n - 1$ ), $s$ is the sample standard deviation, and $n$ is the sample size.

STEP 2

Determine the degrees of freedom for the t-distribution.
$df = n - 1$

STEP 3

Plug in the value of $n$ to find the degrees of freedom.
$df = 8 - 1 = 7$

STEP 4

Determine the value of $\alpha$ (alpha), which is the probability in the tails (the complement of the confidence level).
$\alpha = 1 - c$

STEP 5

Plug in the value of $c$ to find $\alpha$ .
$\alpha = 1 - 0.95 = 0.05$

STEP 6

Since the t-distribution is symmetric, the probability in each tail is $\frac{\alpha}{2}$ .
$\frac{\alpha}{2} = \frac{0.05}{2} = 0.025$

STEP 7

Find the t-score ( $t_{\frac{\alpha}{2}}$ ) that corresponds to the given confidence level and degrees of freedom. This is typically done using a t-distribution table or a calculator with statistical functions.

STEP 8

For a confidence level of 0.95 and 7 degrees of freedom, the t-score is approximately 2.365 (this value may vary slightly depending on the t-distribution table or calculator used).

STEP 9

Now that we have the t-score, we can calculate the margin of error using the formula.
$E = t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}}$

STEP 10

Plug in the values for $t_{\frac{\alpha}{2}}$ , $s$ , and $n$ to calculate the margin of error.
$E = 2.365 \cdot \frac{4}{\sqrt{8}}$

STEP 11

Calculate the square root of $n$ .
$\sqrt{8} = 2.8284$

STEP 12

Now, divide the sample standard deviation by the square root of the sample size.
$\frac{4}{2.8284} \approx 1.4142$

STEP 13

Multiply the t-score by the result from the previous step to find the margin of error.
$E = 2.365 \cdot 1.4142$

STEP 14

Calculate the margin of error.
$E \approx 2.365 \cdot 1.4142 \approx 3.344$
The margin of error for the given values of $c$ , $s$ , and $n$ is approximately 3.344.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms