To find the interval in which Susan can be 68% sure that the sample mean will lie, she needs to calculate the 68% confidence interval for the sample mean. Here are the detailed steps:
1. Determine the Known Values:
- Population Mean (μ): [tex]$98.75$[/tex]
- Population Standard Deviation (σ): [tex]$10.45$[/tex]
- Sample Size (n): [tex]$60$[/tex]
2. Calculate the Standard Error of the Mean (SEM):
- The standard error of the mean is calculated using the formula:
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Plugging in the values:
[tex]\[
\text{SEM} = \frac{10.45}{\sqrt{60}} \approx 1.3490891989289167
\][/tex]
3. Determine the Z-score for a 68% Confidence Interval:
- For a 68% confidence interval, the Z-score is 1 (since one standard deviation from the mean in a normal distribution encompasses approximately 68% of the data).
4. Calculate the Margin of Error (ME):
- The margin of error is calculated using the formula:
[tex]\[
\text{ME} = Z \times \text{SEM}
\][/tex]
- With [tex]\( Z = 1 \)[/tex] and [tex]\( \text{SEM} \approx 1.3490891989289167 \)[/tex]:
[tex]\[
\text{ME} = 1 \times 1.3490891989289167 \approx 1.3490891989289167
\][/tex]
5. Calculate the Confidence Interval:
- The confidence interval is given by:
[tex]\[
\text{Lower Bound} = \mu - \text{ME}
\][/tex]
[tex]\[
\text{Upper Bound} = \mu + \text{ME}
\][/tex]
- Plugging in the values:
[tex]\[
\text{Lower Bound} = 98.75 - 1.3490891989289167 \approx 97.40091080107108
\][/tex]
[tex]\[
\text{Upper Bound} = 98.75 + 1.3490891989289167 \approx 100.09908919892892
\][/tex]
Therefore, the equation and values that Susan can use to find the interval in which she can be 68% sure that the sample mean will lie are:
[tex]\[
97.40091080107108 \leq \text{Sample Mean} \leq 100.09908919892892
\][/tex]
These bounds show the range within which the sample mean is expected to lie with 68% confidence.
