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### Interval Calculation Formula:

We are asked to find the 95% confidence interval for the mean using the formula:
\[
\left[\mu - z\frac{\sigma}{\sqrt{n}}, \mu + z\frac{\sigma}{\sqrt{n}}\right]
\]
Where:
- \(\mu\) is the population mean,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size,
- \(z\) is the z-value corresponding to the 95% confidence level, which is approximately 1.96.

### a) Given:
\[
\mu = 163, \sigma = 17, n = 49
\]
Substitute the values into the formula to calculate the interval.

### b) Given:
\[
\mu = 1014, \sigma = 18, n = 5
\]
Substitute these values to calculate the interval.

### c) Given:
\[
\mu = 56, \sigma = 2, n = 20
\]
We'll compute the confidence interval for this as well. Let me proceed with the calculations.The 95% confidence intervals for each case are as follows:

- **a)** For \(\mu = 163\), \(\sigma = 17\), \(n = 49\):
  \[
  \text{The 95% confidence interval is from } 158.24 \text{ to } 167.76.
  \]

- **b)** For \(\mu = 1014\), \(\sigma = 18\), \(n = 5\):
  \[
  \text{The 95% confidence interval is from } 998.22 \text{ to } 1029.78.
  \]

- **c)** For \(\mu = 56\), \(\sigma = 2\), \(n = 20\):
  \[
  \text{The 95% confidence interval is from } 55.12 \text{ to } 56.88.
  \]

Would you like a more detailed explanation of the calculations or have any further questions?

Here are 5 related questions to explore:
1. What is the significance of the 95% confidence interval in hypothesis testing?
2. How would changing the sample size \(n\) affect the width of the confidence interval?
3. Can confidence intervals be used for non-normal distributions?
4. What is the role of the standard deviation \(\sigma\) in determining the confidence interval?
5. Why is the z-value approximately 1.96 for a 95% confidence interval?

**Tip:** The larger the sample size, the narrower the confidence interval, as the margin of error decreases with increasing \(n\).

Are the following statements true or false?
a. If we see a standardized z-value beyond ±3, the variable cannot be normally distributed.
b. If X and Y are two normally distributed random variables measured in different units (e.g., X is in pounds and Y is in kilograms), then it is not meaningful to compare the standardized z-values.
c. Two machines fill 2-liter soft drink bottles using a similar process. Machine A has μ = 1990ml and σ = 5ml, while Machine B has μ = 1995ml and σ = 3ml. The variables cannot both be normally distributed because they have different standard deviations.

Learn how to calculate confidence intervals using z-scores and understand normal distribution with practical examples. Explore scenarios involving sample means, standard deviations, and how to assess data normality.

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