2. Find the $$ 90 \% $$ confidence interval if there are 85 employees taken the same work with a mean of 180 and standard deviation of 40.

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UpStudy AI · Accepted Answer

To find the $$ 90\% $$ confidence interval for the mean, we can use the formula for the confidence interval:

$$
CI = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
$$

Where:
- $$ \bar{x} $$ is the sample mean,
- $$ z $$ is the z-score corresponding to the desired confidence level,
- $$ s $$ is the sample standard deviation,
- $$ n $$ is the sample size.

### Step 1: Identify the known values
- Sample mean ($$ \bar{x} $$) = 180
- Sample standard deviation ($$ s $$) = 40
- Sample size ($$ n $$) = 85

### Step 2: Find the z-score for a $$ 90\% $$ confidence level
For a $$ 90\% $$ confidence level, the z-score can be found using a standard normal distribution table or calculator. The z-score for $$ 90\% $$ confidence is approximately $$ 1.645 $$.

### Step 3: Calculate the standard error (SE)
The standard error (SE) is calculated as follows:

$$
SE = \frac{s}{\sqrt{n}} = \frac{40}{\sqrt{85}}
$$

### Step 4: Calculate the confidence interval
Now we can calculate the confidence interval using the values we have:

$$
CI = 180 \pm 1.645 \left( \frac{40}{\sqrt{85}} \right)
$$

Let's perform the calculations for the standard error and the confidence interval.
Calculate the value by following steps:
- step0: Calculate:
  $$180+1.645\left(\frac{40}{\sqrt{85}}\right)$$
- step1: Remove the parentheses:
  $$180+1.645\times \frac{40}{\sqrt{85}}$$
- step2: Multiply the numbers:
  $$180+\frac{329}{5\sqrt{85}}$$
- step3: Calculate:
  $$180+\frac{329\sqrt{85}}{425}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{180\times 425}{425}+\frac{329\sqrt{85}}{425}$$
- step5: Transform the expression:
  $$\frac{180\times 425+329\sqrt{85}}{425}$$
- step6: Multiply the numbers:
  $$\frac{76500+329\sqrt{85}}{425}$$
Calculate or simplify the expression $$ 180 - 1.645 * (40/\sqrt(85)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$180-1.645\left(\frac{40}{\sqrt{85}}\right)$$
- step1: Remove the parentheses:
  $$180-1.645\times \frac{40}{\sqrt{85}}$$
- step2: Multiply the numbers:
  $$180-\frac{329}{5\sqrt{85}}$$
- step3: Calculate:
  $$180-\frac{329\sqrt{85}}{425}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{180\times 425}{425}-\frac{329\sqrt{85}}{425}$$
- step5: Transform the expression:
  $$\frac{180\times 425-329\sqrt{85}}{425}$$
- step6: Multiply the numbers:
  $$\frac{76500-329\sqrt{85}}{425}$$
Calculate or simplify the expression $$ 40/\sqrt(85) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{40}{\sqrt{85}}$$
- step1: Multiply by the Conjugate:
  $$\frac{40\sqrt{85}}{\sqrt{85}\times \sqrt{85}}$$
- step2: Multiply the numbers:
  $$\frac{40\sqrt{85}}{85}$$
- step3: Reduce the fraction:
  $$\frac{8\sqrt{85}}{17}$$
### Step 3: Calculate the Standard Error (SE)
The standard error (SE) is calculated as follows:

$$
SE \approx 4.34
$$

### Step 4: Calculate the Confidence Interval
Now we can calculate the confidence interval:

1. **Upper limit**:
   $$
   CI_{upper} \approx 180 + 1.645 \times 4.34 \approx 187.14
   $$

2. **Lower limit**:
   $$
   CI_{lower} \approx 180 - 1.645 \times 4.34 \approx 172.86
   $$

### Final Confidence Interval
Thus, the $$ 90\% $$ confidence interval for the mean is:

$$
(172.86, 187.14)
$$

This means we are $$ 90\% $$ confident that the true mean of the population lies between $$ 172.86 $$ and $$ 187.14 $$.
The $$ 90\% $$ confidence interval for the mean is approximately $$ 172.86 $$ to $$ 187.14 $$.

2. Find the \( 90 \% \) confidence interval if there are 85 employees taken the same work with a mean of 180 and standard deviation of 40.

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