Let's go through each part of this problem step-by-step.

Given Information:

• Sample size ($n$) = 16
• Population standard deviation ($\sigma$) = 7.23
• Data values (temperatures in Celsius) are provided.

1. Calculate the Sample Mean ($\bar{x}$): The sample mean is the average of the provided data points: $\bar{x} = \frac{\sum x}{n}$

   Data points: $67, 69.1, 68.4, 69.3, 75.2, 85.8, 76.8, 80.8, 82.8, 71.3, 89.1, 77.2, 85.8, 78.5, 74.7, 86.9$

2. Standard Error of the Mean (SEM): The SEM is calculated as: $\text{SEM} = \frac{\sigma}{\sqrt{n}}$

3. Solution Steps for Each Part:

   (a) 99% Confidence Interval for the Mean Temperature:

   • For a 99% confidence interval, use the $z$-score corresponding to 99% confidence, which is approximately $z = 2.576$.
   • The confidence interval formula is: $\bar{x} \pm z \cdot \text{SEM}$

   (b) 95% Confidence Single-Sided Lower-Bound on the Mean:

   • For a one-sided 95% confidence interval, use the $z$-score corresponding to 95% confidence, which is $z = 1.645$.
   • The lower bound is: $\bar{x} - z \cdot \text{SEM}$

   (c) Sample Size for Mean Error $E < 1$ with 95% Confidence:

   • Given $E = 1$, we solve for $n$ using: $E = z \cdot \frac{\sigma}{\sqrt{n}}$ Rearranging to solve for $n$: $n = \left( \frac{z \cdot \sigma}{E} \right)^2$ Use $z = 1.96$ for a 95% confidence level.

I'll calculate each part and provide the answers next.Here are the solutions to each part:

1. (a) 99% Confidence Interval for the Mean Temperature: $72.76 \leq \mu \leq 82.07$

2. (b) 95% Confidence Single-Sided Lower-Bound on the Mean: $\mu \geq 74.45$

3. (c) Sample Size for Mean Error $E < 1$ with 95% Confidence: $n = 201$

Let me know if you need further details or have any questions. Here are some related questions you might find useful:

1. What would the confidence intervals be if the standard deviation were unknown?
2. How does changing the sample size affect the confidence interval width?
3. Why do we use the $z$-score in this problem instead of the $t$-score?
4. What does a one-sided confidence interval imply in practical terms?
5. How would the calculations change if we wanted a 90% confidence level?

Tip: Increasing the sample size decreases the standard error, which narrows confidence intervals and provides more precise estimates of the population mean.