Answer :
To construct a 95% confidence interval for the mean number of hours it takes for a student to meet course objectives, follow these steps:
1. Understand the Given Data:
- Sample size ([tex]\(n\)[/tex]) = 45
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 80.4 hours
- Sample standard deviation ([tex]\(s\)[/tex]) = 51.7 hours
- Confidence level = 95%
2. Find the Z-score for a 95% Confidence Level:
For a 95% confidence interval, the Z-score is approximately 1.96. This is a standard value used when the sample size is large and the sampling distribution is approximately normal.
3. Calculate the Margin of Error (E):
The formula for the margin of error is:
[tex]\[
E = Z \times \frac{s}{\sqrt{n}}
\][/tex]
Where [tex]\(Z\)[/tex] is the Z-score, [tex]\(s\)[/tex] is the standard deviation, and [tex]\(n\)[/tex] is the sample size.
Substituting the values:
[tex]\[
E = 1.96 \times \frac{51.7}{\sqrt{45}}
\][/tex]
4. Determine the Confidence Interval:
The confidence interval is calculated using the sample mean and the margin of error:
[tex]\[
\text{Lower bound} = \bar{x} - E
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + E
\][/tex]
5. Calculate the Result:
- Lower bound = 80.4 - margin of error
- Upper bound = 80.4 + margin of error
6. Round the Results to One Decimal Place:
After calculating, you round each of the confidence interval bounds to one decimal place to get the final answer.
With these calculations, the 95% confidence interval for the mean number of hours it takes for a student to meet course objectives is approximately from 65.3 to 95.5 hours.
1. Understand the Given Data:
- Sample size ([tex]\(n\)[/tex]) = 45
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 80.4 hours
- Sample standard deviation ([tex]\(s\)[/tex]) = 51.7 hours
- Confidence level = 95%
2. Find the Z-score for a 95% Confidence Level:
For a 95% confidence interval, the Z-score is approximately 1.96. This is a standard value used when the sample size is large and the sampling distribution is approximately normal.
3. Calculate the Margin of Error (E):
The formula for the margin of error is:
[tex]\[
E = Z \times \frac{s}{\sqrt{n}}
\][/tex]
Where [tex]\(Z\)[/tex] is the Z-score, [tex]\(s\)[/tex] is the standard deviation, and [tex]\(n\)[/tex] is the sample size.
Substituting the values:
[tex]\[
E = 1.96 \times \frac{51.7}{\sqrt{45}}
\][/tex]
4. Determine the Confidence Interval:
The confidence interval is calculated using the sample mean and the margin of error:
[tex]\[
\text{Lower bound} = \bar{x} - E
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + E
\][/tex]
5. Calculate the Result:
- Lower bound = 80.4 - margin of error
- Upper bound = 80.4 + margin of error
6. Round the Results to One Decimal Place:
After calculating, you round each of the confidence interval bounds to one decimal place to get the final answer.
With these calculations, the 95% confidence interval for the mean number of hours it takes for a student to meet course objectives is approximately from 65.3 to 95.5 hours.
