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To determine if the data from the jars of peanuts follow a normal distribution, we use a tool called a normal probability plot. This plot helps you visualize the data points against a theoretical normal distribution. If the data points align roughly on a straight line, then we can say the data are normally distributed. To create this plot, you can use statistical software or a graphing calculator. Input your data points (15.94, 15.74, 16.21, 15.36, 15.84, 15.84, 15.52, 16.16, 15.78, 15.51, 16.28, 16.53) and generate the plot. If the resulting graph shows that the data points lie close to a straight line, it confirms the assumption of normality. Understanding if your data is normally distributed is crucial for many statistical methods and tests.

The sample standard deviation is a measure of the amount of variation or dispersion in a set of values. To calculate it, follow these steps:

• Step 1: Find the mean (average) of your data set using the formula:
  
  \( \overline{x} = \frac{\text{sum of all data points}}{n} \)
  
  For our data, it will be: \( \overline{x} = \frac{15.94 + 15.74 + 16.21 + 15.36 + 15.84 + 15.84 + 15.52 + 16.16 + 15.78 + 15.51 + 16.28 + 16.53}{12} \)
• Step 2: Subtract the mean from each data point to find the deviation of each point.
• Step 3: Square each deviation.
• Step 4: Sum all the squared deviations.
• Step 5: Divide this sum by the sample size minus one (n-1) to find the sample variance.
• Step 6: Finally, take the square root of the variance to get the sample standard deviation (s).
  
  \( s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2 } \)

For our data set, this step-by-step method will provide us with the variability in the peanut jar weights.

A confidence interval gives us a range of values which is likely to contain an unknown population parameter, such as the population standard deviation. For this problem, we want a 90% confidence interval for the population standard deviation of the number of ounces of peanuts. To construct this interval, we use the chi-square distribution. Here's the formula:

\( \left(\frac{(n-1)s^2}{\chi^2_{(1-\frac{\alpha}{2},n-1)}}, \frac{(n-1)s^2}{\chi^2_{(\frac{\alpha}{2},n-1)}}\right) \)

Where:

• n is the sample size (12).
• s is the sample standard deviation we calculated.
• \( \chi^2 \) is the chi-square critical value corresponding to our confidence level (in this case, 90%).
• \( \alpha = 1 - 0.90 = 0.10 \)

By plugging in these values and using chi-square tables or software for critical values, we get the appropriate confidence interval. This interval provides a range where we expect the population standard deviation to lie with 90% confidence. It's an essential tool for understanding the variability in our process and determining if the machine is functioning correctly.

The population standard deviation is a measure of the spread of all possible values of a dataset. Unlike the sample standard deviation which we calculated from our 12 jars, the population standard deviation accounts for the entire population or production from the filling machine. For this problem, the manager desires a population standard deviation lower than 0.20 ounces.

After constructing our 90% confidence interval, we compare it with the desired 0.20 ounces. If the entire confidence interval is below 0.20 ounces, it indicates that the machine's variability meets the manager's standards. However, if 0.20 ounces is within the interval, then we cannot be confident that the machine meets this requirement.

Monitoring the population standard deviation is crucial in quality control, as it helps to maintain consistent product weight, ensuring both customer satisfaction and regulatory compliance.