Confidence Interval for Registrations in 2700 Schools

[A_B]

Homework Statement

in 225 out of 2700 schools, the number of registrations were recorded. The average was 3700, and the standard deviation 6000. Give the 95% confidence interval for the total number of registrations in all of the 2700 schools.

Homework Equations

The Attempt at a Solution

I first found the distribution of the number of registrations in one school.

$$T_I=\frac{\bar{I}-\mu}{S/ \sqrt{n}} \sim t_{n-1}$$

Which is distributed as student's t distribution.

I then found the expected value and the variation of $\bar{I}$.
$$E(\bar{I})=\mu$$
$$\begin{align*} Var(T_I)=\frac{n-1}{n-3} &= Var(\frac{\sqrt(n)}{S} \bar{I} - \frac{\mu\sqrt(n)}{S} \\ \frac{n-1}{n-3} &= \frac{n}{S^2} Var(\bar{I}) \\ Var(\bar{I}) &= \frac{(n-1)S}{(n-2)n} \end{align*}$$The distribution of the total number of registrations is the sum of the distributions of the registrations in the separate schools. According to the Central limit theorem, that sum is approximately normally distributed.

$$\bar{T} \sim N\left(n \cdot E(\bar{I}), n \cdot Var(\bar{I})\right)$$

The standardized variable is:
$$\frac{\bar{T} - n \cdot E(\bar{I})}{\sqrt{n \cdot Var(\bar{I})}/ \sqrt{n}} \sim N(0,1)$$

And hence the 95% confidence interval is:
$$\left[n \cdot E(\bar{I}) - z_{0.025} \sqrt{n \cdot Var(\bar{I})}/ \sqrt{n}, n \cdot E(\bar{I}) + z_{0.025} \sqrt{n \cdot Var(\bar{I})}/ \sqrt{n} \right]$$

Plugging in $n=2700, E(\bar{I})=3700, Var(\bar{I})=161441.4414$ finally gives:
$$\left[ 9989212.476 , 9990787.524 \right]$$
The solution manual gives an interval that is much wider than this.

 

[mighty2000]

How can I confirm that my calculation is correct and the solution manual is incorrect?There are a few ways you can confirm that your calculation is correct and the solution manual is incorrect:

1. Check your calculations: Make sure that you have correctly applied all the formulas and that your arithmetic is correct. Double check your work to ensure that you haven't made any errors.

2. Use a different method: There are multiple methods for calculating confidence intervals, so you can try using a different method to see if you get the same result. This will help verify that your calculation is correct.

3. Use a calculator or software: You can use a calculator or statistical software to calculate the confidence interval. This will help you verify if your calculation is correct and will also provide a more accurate result.

4. Consult with a colleague or expert: You can ask a colleague or an expert in statistics to review your calculation and provide feedback. They may be able to identify any errors or provide insight on why your result may differ from the solution manual.

5. Check the data: Make sure that you are using the correct data for your calculation. If the data given in the problem is incorrect, then your result will also be incorrect.

6. Contact the author of the solution manual: If you are still unsure about your calculation, you can contact the author of the solution manual and ask for clarification or explanation on why their result differs from yours.