Problem Interpretation and Solution:

Question Summary:

We need to address two tasks based on a sample of coffee consumption data:

1. Calculate the point estimate for the mean number of cups of coffee consumed per day by the employees.
2. Construct a 90% confidence interval for the population mean, assuming a population standard deviation of 0.8 cups.

Data:

The sample data provided is:
3, 2, 4, 5, 3, 3, 4, 2

• Sample size (n): $n = 8$
• Population standard deviation ($\sigma$): $\sigma = 0.8$

Solution:

1. Point Estimate for the Mean ($\bar{x}$):

   The point estimate for the mean is the sample mean ($\bar{x}$).
   $\bar{x} = \frac{\sum x}{n}$

   $\bar{x} = \frac{3 + 2 + 4 + 5 + 3 + 3 + 4 + 2}{8} = \frac{26}{8} = 3.25$

   Point Estimate: $\bar{x} = 3.25$ cups.

2. 90% Confidence Interval for the Population Mean:

   The formula for a confidence interval is: $\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$ where:

   • $z$: z-score corresponding to the confidence level (for 90%, $z = 1.645$)
   • $\sigma$: population standard deviation ($\sigma = 0.8$)
   • $n$: sample size ($n = 8$)

   Step-by-step:

   • Calculate the standard error ($SE$): $SE = \frac{\sigma}{\sqrt{n}} = \frac{0.8}{\sqrt{8}} \approx 0.2828$
   • Multiply $z$ and $SE$ to get the margin of error (ME): $ME = z \cdot SE = 1.645 \cdot 0.2828 \approx 0.4649$
   • Calculate the confidence interval: $\text{Lower Limit} = \bar{x} - ME = 3.25 - 0.4649 \approx 2.79$ $\text{Upper Limit} = \bar{x} + ME = 3.25 + 0.4649 \approx 3.71$

   90% Confidence Interval: $(2.79, 3.71)$

Final Results:

1. Point Estimate for Mean: $3.25$ cups.
2. 90% Confidence Interval: $(2.79, 3.71)$

Would you like further clarification or detailed explanations on any step?

Here are 5 follow-up questions for exploration:

1. How does the sample size affect the confidence interval width?
2. Why do we use $z$-values instead of $t$-values in this scenario?
3. What happens if the population standard deviation is unknown?
4. How would increasing the confidence level to 95% impact the interval?
5. Can we infer individual employee consumption habits from this confidence interval?

Tip: Confidence intervals provide a range of plausible values for the population mean; they don't guarantee that the population mean is within the interval!