Math / Data & Statistics

Questionwww-awu.aleks.com tting and Onlin.... Sports Betting Secret - 11/22/24 ATL @ CHI III St... Home - Northern Essex... Content ChatGPT (5) KaiCenat - Twitcr Homework \# 4: $7(1,2,3,4) 8(2,3,4)$ Question 40 of 40 (1 point) I Question Attempt: 1 of 3 Jonathan Español
Eat your cereal: Boxes of cereal are labeled as containing 16 ounces. Following are the weights of a sample of 12 boxes. Assume the population is normally distributed. \begin{tabular}{llllll} \hline 16.05 & 15.96 & 16.13 & 16.07 & 16.17 & 16.07 \\ 16.03 & 16.17 & 16.16 & 15.96 & 16.18 & 16.18 \\ \hline \end{tabular} Send data to Excel
Part 1 of 3 (a) Find the sample standard deviation. Round the answer to at least four decimal places.
The sample standard deviation $s$ is 0.0452 .
Correct Answer: $0.0826$
Part: $1 / 3$
Part 2 of 3 (b) Construct a $99 \%$ confidence interval for the population standard deviation $\sigma$ . Round the answers to at least two decimal places.
A $99 \%$ confidence interval for the population standard deviation is $\square$ $<\sigma<$ $\square$ . Skip Part Check Save For Later Submit Assign

Studdy Solution

STEP 1

What is this asking? We need to calculate the 99% confidence interval for the standard deviation of cereal box weights, given a sample of 12 boxes. Watch out! Don't confuse sample standard deviation (s) with population standard deviation (σ).
Also, remember the correct formula for the confidence interval of a standard deviation uses the Chi-Square distribution!

STEP 2

1. Calculate Sample Standard Deviation
2. Calculate Degrees of Freedom
3. Find Chi-Square Values
4. Calculate Confidence Interval

STEP 3

First, let's calculate the sample mean $\bar{x}$ .
Add up all the weights and divide by the number of boxes, which is 12:
$\bar{x} = \frac{16.05 + 15.96 + 16.13 + 16.07 + 16.17 + 16.07 + 16.03 + 16.17 + 16.16 + 15.96 + 16.18 + 16.18}{12}$ $\bar{x} = \frac{192.3}{12}$ $\bar{x} = 16.025$

STEP 4

Now, we'll calculate the sum of squared differences from the mean.
This means subtracting the mean from each weight, squaring the result, and adding all those squared differences together.
$\sum_{i=1}^{12} (x_i - \bar{x})^2 = (16.05 - 16.025)^2 + (15.96 - 16.025)^2 + ... + (16.18 - 16.025)^2$ $= 0.000625 + 0.004225 + ... + 0.024025$ $= 0.0756$

STEP 5

To get the sample variance, divide the sum of squared differences by $n-1$ , which is 11 (one less than the sample size).
$s^2 = \frac{0.0756}{11}$ $s^2 = 0.00687272...$

STEP 6

Finally, take the square root of the sample variance to find the sample standard deviation s:
$s = \sqrt{0.00687272...}$ $s \approx 0.0829$

STEP 7

The degrees of freedom (df) is simply the sample size minus 1.
Since we have 12 boxes, our degrees of freedom is:
$df = 12 - 1 = 11$

STEP 8

For a 99% confidence interval with 11 degrees of freedom, we need to find two Chi-Square values from a Chi-Square table or calculator.
We want the values that cut off 0.5% on each tail, so we're looking for $\chi^2_{0.005}$ and $\chi^2_{0.995}$ .
These values are:
$\chi^2_{0.005} = 26.757$ $\chi^2_{0.995} = 2.603$

STEP 9

Now, we'll plug everything into the confidence interval formula for the standard deviation:
$\sqrt{\frac{(n-1) \cdot s^2}{\chi^2_{\alpha/2}}} < \sigma < \sqrt{\frac{(n-1) \cdot s^2}{\chi^2_{1-\alpha/2}}}$

STEP 10

Substitute the values we've found: n = 12, s ≈ 0.0829, $\chi^2_{0.005}$ = 26.757, and $\chi^2_{0.995}$ = 2.603.
$\sqrt{\frac{(12-1) \cdot (0.0829)^2}{26.757}} < \sigma < \sqrt{\frac{(12-1) \cdot (0.0829)^2}{2.603}}$ $\sqrt{\frac{11 \cdot 0.006872}{26.757}} < \sigma < \sqrt{\frac{11 \cdot 0.006872}{2.603}}$ $\sqrt{0.00278} < \sigma < \sqrt{0.0288}$ $0.053 < \sigma < 0.170$

STEP 11

The 99% confidence interval for the population standard deviation σ is approximately **0.053 < σ < 0.170**.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Sample Standard Deviation
2. Calculate Degrees of Freedom
3. Find Chi-Square Values
4. Calculate Confidence Interval

STEP 3

First, let's calculate the sample mean $\bar{x}$ .
Add up all the weights and divide by the number of boxes, which is 12:
$\bar{x} = \frac{16.05 + 15.96 + 16.13 + 16.07 + 16.17 + 16.07 + 16.03 + 16.17 + 16.16 + 15.96 + 16.18 + 16.18}{12}$ $\bar{x} = \frac{192.3}{12}$ $\bar{x} = 16.025$

STEP 4

STEP 5

To get the sample variance, divide the sum of squared differences by $n-1$ , which is 11 (one less than the sample size).
$s^2 = \frac{0.0756}{11}$ $s^2 = 0.00687272...$

STEP 6

Finally, take the square root of the sample variance to find the sample standard deviation s:
$s = \sqrt{0.00687272...}$ $s \approx 0.0829$

STEP 7

The degrees of freedom (df) is simply the sample size minus 1.
Since we have 12 boxes, our degrees of freedom is:
$df = 12 - 1 = 11$

STEP 8

STEP 9

Now, we'll plug everything into the confidence interval formula for the standard deviation:
$\sqrt{\frac{(n-1) \cdot s^2}{\chi^2_{\alpha/2}}} < \sigma < \sqrt{\frac{(n-1) \cdot s^2}{\chi^2_{1-\alpha/2}}}$

STEP 10

STEP 11

The 99% confidence interval for the population standard deviation σ is approximately **0.053 < σ < 0.170**.

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Studdy Solution

STEP 1

STEP 2

1. Calculate Sample Standard Deviation2. Calculate Degrees of Freedom3. Find Chi-Square Values4. Calculate Confidence Interval

STEP 3

STEP 4

STEP 5

To get the **sample variance**, divide the sum of squared differences by n−1n-1n−1, which is **11** (one less than the sample size).s2=0.075611 s^2 = \frac{0.0756}{11} s2=110.0756​s2=0.00687272... s^2 = 0.00687272... s2=0.00687272...

STEP 6

Finally, take the **square root of the sample variance** to find the **sample standard deviation** *s*:s=0.00687272... s = \sqrt{0.00687272...} s=0.00687272...​s≈0.0829 s \approx 0.0829 s≈0.0829

STEP 7

The **degrees of freedom** (df) is simply the sample size minus 1.Since we have **12** boxes, our degrees of freedom is:df=12−1=11 df = 12 - 1 = 11 df=12−1=11

STEP 8

STEP 9

STEP 10

STEP 11

The 99% confidence interval for the population standard deviation *σ* is approximately **0.053 < *σ* < 0.170**.

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

1. Calculate Sample Standard Deviation
2. Calculate Degrees of Freedom
3. Find Chi-Square Values
4. Calculate Confidence Interval

To get the sample variance, divide the sum of squared differences by $n-1$ , which is 11 (one less than the sample size).
$s^2 = \frac{0.0756}{11}$ $s^2 = 0.00687272...$

Finally, take the square root of the sample variance to find the sample standard deviation s:
$s = \sqrt{0.00687272...}$ $s \approx 0.0829$

The degrees of freedom (df) is simply the sample size minus 1.
Since we have 12 boxes, our degrees of freedom is:
$df = 12 - 1 = 11$

The 99% confidence interval for the population standard deviation σ is approximately **0.053 < σ < 0.170**.