Solved on Dec 05, 2023

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

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.

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

STEP 1

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.

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

STEP 1

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

Plug in the value of ccc to find α\alphaα.α=1−0.95=0.05 \alpha = 1 - 0.95 = 0.05 α=1−0.95=0.05

STEP 6

Since the t-distribution is symmetric, the probability in each tail is α2\frac{\alpha}{2}2α​.α2=0.052=0.025 \frac{\alpha}{2} = \frac{0.05}{2} = 0.025 2α​=20.05​=0.025

STEP 7

Find the t-score (tα2t_{\frac{\alpha}{2}}t2α​​) 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=tα2⋅sn E = t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}} E=t2α​​⋅n​s​

STEP 10

Plug in the values for tα2t_{\frac{\alpha}{2}}t2α​​, sss, and nnn to calculate the margin of error.E=2.365⋅48 E = 2.365 \cdot \frac{4}{\sqrt{8}} E=2.365⋅8​4​

STEP 11

Calculate the square root of nnn.8=2.8284 \sqrt{8} = 2.8284 8​=2.8284

STEP 12

Now, divide the sample standard deviation by the square root of the sample size.42.8284≈1.4142 \frac{4}{2.8284} \approx 1.4142 2.82844​≈1.4142

STEP 13

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

STEP 14

Calculate the margin of error.E≈2.365⋅1.4142≈3.344 E \approx 2.365 \cdot 1.4142 \approx 3.344 E≈2.365⋅1.4142≈3.344The margin of error for the given values of ccc, sss, and nnn is approximately 3.344.

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

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

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

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

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

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

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.

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

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

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

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

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

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.