She obtains the accompanying data Complete parts (a) through (d) below (a) Venty that the data are normally distributed by constructed a normal probability plot. Choose the correct normal probability plot below OA О в. OC. & expected 2-score 1.754 -1.750 11.2 O No O 12.6 Are the data normally distributed? Yes a (b) Determine the sample standard deviation s (Round to three decimal places as needed) A expected zocoro 126 11.2 -1.75 129 observation Q G 1.75 expected 2-score 1.754 112 observation OD. 4 expected score 175 112 69978447 observation (c) Construct a 99% confidence interval for the population standard deviation of the number of ounces of peanuts Select the correct choice below and fill in the answer boxes to complete your choice (Use ascending order Round to three decimal places as needed) OA. There is a 99% chance that the true population standard deviation is between and 126 Q

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Peanut Jar Data:

11.94

11.74

12.25

11.32

11.85

11.86

11.59

12.15

11.76

11.55

12.31

12.59

(c) Construct a 99% confidence interval for the population standard deviation of the number of ounces of peanuts. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to three decimal places as needed.)
- A. There is a 99% chance that the true population standard deviation is between [ ] and [ ].
- B. If repeated samples are taken, 99% of them will have the sample standard deviation between [ ] and [ ].
- C. There is 99% confidence that the population standard deviation is between [ ] and [ ].

(d) The quality control manager wants the machine to have a population standard deviation below 0.20 ounce. Does the confidence interval validate this desire?
- A. No—the lower bound of the confidence interval is less than 0.20.
- B. Yes—the lower bound of the confidence interval is less than 0.20.
- C. No—the lower bound of the confidence interval is greater than 0.20.
- D. Yes—the lower bound of the confidence interval is greater than 0.20.
Transcribed Image Text:(c) Construct a 99% confidence interval for the population standard deviation of the number of ounces of peanuts. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to three decimal places as needed.) - A. There is a 99% chance that the true population standard deviation is between [ ] and [ ]. - B. If repeated samples are taken, 99% of them will have the sample standard deviation between [ ] and [ ]. - C. There is 99% confidence that the population standard deviation is between [ ] and [ ]. (d) The quality control manager wants the machine to have a population standard deviation below 0.20 ounce. Does the confidence interval validate this desire? - A. No—the lower bound of the confidence interval is less than 0.20. - B. Yes—the lower bound of the confidence interval is less than 0.20. - C. No—the lower bound of the confidence interval is greater than 0.20. - D. Yes—the lower bound of the confidence interval is greater than 0.20.
### Educational Content: Understanding Normal Distribution with a Probability Plot

**Problem Overview:**
A jar of peanuts is supposed to contain 12 ounces. However, with inevitable fluctuations in the filling process, a quality-control manager randomly samples 12 jars and measures their contents. The task is to analyze whether the data is normally distributed, calculate the sample standard deviation, and construct a 99% confidence interval for the population standard deviation.

---

**Part (a): Verification of Normal Distribution**

To verify if the data is normally distributed, a normal probability plot is used. The correct plot should show an approximate linear relationship between the expected z-scores and the observed data. Out of four options (A, B, C, D), choose the plot that demonstrates this linearity, confirming normality.

**Options:**

- **Plot A**: Displays a scattered relationship, indicating non-normal distribution.
- **Plot B**: Shows a clear linear trend.
- **Plot C**: Demonstrates some clustering, suggesting deviation from normality.
- **Plot D**: Scattered with no clear linearity.

**Question:** Are the data normally distributed?
- [ ] No
- [ ] Yes

**Correct Answer:** Plot B shows the most linearity, suggesting the data are normally distributed.

---

**Part (b): Determination of Sample Standard Deviation**

Calculate the sample standard deviation (s) using the observed data, rounding to three decimal places as needed.

**Formula:**
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]

---

**Part (c): Confidence Interval for Population Standard Deviation**

Construct a 99% confidence interval for the population standard deviation based on the sample data. Select the correct range:

- **Option A**: Suggests a 99% chance that the true population standard deviation is between two values (to be calculated and inserted in the blanks).
  
- **Option B**: Refers to repeated samples, indicating 99% of them will have a sample standard deviation within a specified range (calculated and filled in the blanks).

**Correct Approach:** Use statistical methods (such as a chi-square distribution) to find the interval and fill in the blanks.

---

**Conclusion:**

This exercise demonstrates practical methods of assessing normality in data, calculating standard deviation, and constructing confidence intervals. Understanding these concepts is crucial for quality control and statistical analysis in manufacturing processes.
Transcribed Image Text:### Educational Content: Understanding Normal Distribution with a Probability Plot **Problem Overview:** A jar of peanuts is supposed to contain 12 ounces. However, with inevitable fluctuations in the filling process, a quality-control manager randomly samples 12 jars and measures their contents. The task is to analyze whether the data is normally distributed, calculate the sample standard deviation, and construct a 99% confidence interval for the population standard deviation. --- **Part (a): Verification of Normal Distribution** To verify if the data is normally distributed, a normal probability plot is used. The correct plot should show an approximate linear relationship between the expected z-scores and the observed data. Out of four options (A, B, C, D), choose the plot that demonstrates this linearity, confirming normality. **Options:** - **Plot A**: Displays a scattered relationship, indicating non-normal distribution. - **Plot B**: Shows a clear linear trend. - **Plot C**: Demonstrates some clustering, suggesting deviation from normality. - **Plot D**: Scattered with no clear linearity. **Question:** Are the data normally distributed? - [ ] No - [ ] Yes **Correct Answer:** Plot B shows the most linearity, suggesting the data are normally distributed. --- **Part (b): Determination of Sample Standard Deviation** Calculate the sample standard deviation (s) using the observed data, rounding to three decimal places as needed. **Formula:** \[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \] --- **Part (c): Confidence Interval for Population Standard Deviation** Construct a 99% confidence interval for the population standard deviation based on the sample data. Select the correct range: - **Option A**: Suggests a 99% chance that the true population standard deviation is between two values (to be calculated and inserted in the blanks). - **Option B**: Refers to repeated samples, indicating 99% of them will have a sample standard deviation within a specified range (calculated and filled in the blanks). **Correct Approach:** Use statistical methods (such as a chi-square distribution) to find the interval and fill in the blanks. --- **Conclusion:** This exercise demonstrates practical methods of assessing normality in data, calculating standard deviation, and constructing confidence intervals. Understanding these concepts is crucial for quality control and statistical analysis in manufacturing processes.
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The given data is 11.94, 11.74, 12.25, 11.32, 11.85, 11.86, 11.59, 12.15, 11.76, 11.55, 12.31, 12.59



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