Boxes of Honey-Nut Oatmeal are produced to contain 16.0 ounces, with a standard deviation of 0.15 ounce. For a sample size of 49 , the 3 -sigma $$ \bar{x} $$ chart control limits are: Upper Control Limit $$ \left(\right. $$ UCL $$ \left._{\bar{x}}\right)=\square $$ ounces (round your response to two decimal places). Lower Control Limit $$ \left(\right. $$ LCL $$ \left._{\bar{x}}^{-}\right)=\square $$ ounces (round your response to two decimal places).

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To calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL) for the $$ \bar{x} $$ chart, we can use the following formulas:

1. **UCL**: 
   $$
   UCL_{\bar{x}} = \mu + 3 \left( \frac{\sigma}{\sqrt{n}} \right)
   $$

2. **LCL**: 
   $$
   LCL_{\bar{x}} = \mu - 3 \left( \frac{\sigma}{\sqrt{n}} \right)
   $$

Where:
- $$ \mu $$ is the mean (16.0 ounces),
- $$ \sigma $$ is the standard deviation (0.15 ounces),
- $$ n $$ is the sample size (49).

Now, let's calculate the standard error $$ \frac{\sigma}{\sqrt{n}} $$:

$$
\frac{\sigma}{\sqrt{n}} = \frac{0.15}{\sqrt{49}} = \frac{0.15}{7} = 0.02142857
$$

Next, we can calculate the UCL and LCL:

1. **Calculate UCL**:
   $$
   UCL_{\bar{x}} = 16.0 + 3 \times 0.02142857
   $$

2. **Calculate LCL**:
   $$
   LCL_{\bar{x}} = 16.0 - 3 \times 0.02142857
   $$

Now, let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$16-3\times 0.02142857$$
- step1: Multiply the numbers:
  $$16-0.06428571$$
- step2: Subtract the numbers:
  $$15.93571429$$
Calculate or simplify the expression $$ 16.0 + 3 * 0.02142857 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$16+3\times 0.02142857$$
- step1: Multiply the numbers:
  $$16+0.06428571$$
- step2: Add the numbers:
  $$16.06428571$$
Now, let's round the calculated values for the Upper Control Limit (UCL) and Lower Control Limit (LCL) to two decimal places:

1. **UCL**:
   $$
   UCL_{\bar{x}} = 16.06428571 \approx 16.06 \text{ ounces}
   $$

2. **LCL**:
   $$
   LCL_{\bar{x}} = 15.93571429 \approx 15.94 \text{ ounces}
   $$

Thus, the control limits are:
- Upper Control Limit (UCL) $$ = 16.06 $$ ounces
- Lower Control Limit (LCL) $$ = 15.94 $$ ounces
Upper Control Limit (UCL) = 16.06 ounces Lower Control Limit (LCL) = 15.94 ounces

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