To construct a 90% confidence interval for the mean exam score using Excel, follow these steps:

Step 1: Input the Data

1. Enter the given data into a column in Excel. For example, enter the data from cell A1 to A20.

Step 2: Calculate the Mean

1. Use the AVERAGE function to calculate the mean (x̄) of the data.
   • Formula: =AVERAGE(A1:A20)

Step 3: Calculate the Sample Standard Deviation

1. Use the STDEV.S function to calculate the sample standard deviation (s).
   • Formula: =STDEV.S(A1:A20)

Step 4: Determine the Sample Size (n)

1. The sample size (n) is 20 (as there are 20 scores).

Step 5: Find the t-Score for the 90% Confidence Level

1. Use the T.INV.2T function to find the t-score corresponding to a 90% confidence level.
   • Formula: =T.INV.2T(0.10, n-1)
   • In this case: =T.INV.2T(0.10, 19)

Step 6: Calculate the Margin of Error (E)

1. Use the formula $E = t \times \frac{s}{\sqrt{n}}$ to calculate the margin of error.
   • Formula: =t * (s / SQRT(n))

Step 7: Determine the Confidence Interval

1. Calculate the lower bound of the confidence interval:
   • Formula: =Mean - E
2. Calculate the upper bound of the confidence interval:
   • Formula: =Mean + E

Example Calculation in Excel:

Assuming the data is in cells A1 to A20:

1. Mean (x̄): =AVERAGE(A1:A20)
2. Sample Standard Deviation (s): =STDEV.S(A1:A20)
3. t-Score for 90% Confidence Level: =T.INV.2T(0.10, 19) (which gives approximately 1.729)
4. Margin of Error (E): =t * (s / SQRT(20))
5. Lower Bound: =Mean - E
6. Upper Bound: =Mean + E

Final Calculation:

Let’s use these exact numbers:

1. Mean: $\text{Mean} \approx 83.9$
2. Standard Deviation (s): $\text{s} \approx 11.1$
3. t-Score: $\text{t} \approx 1.729$
4. Margin of Error (E): $E \approx 4.3$

Finally, the 90% confidence interval is: $(83.9 - 4.3, 83.9 + 4.3)$

So, the 90% confidence interval for the mean exam score is approximately: $\textbf{(79.6, 88.2)}$

Summary:

The 90% confidence interval for the mean exam score is $\textbf{(79.6, 88.2)}$.

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Would you like a detailed explanation of any of these steps or help with further questions?

Here are five additional questions you might find interesting:

1. How would the confidence interval change if the sample size were increased to 30?
2. What would be the 95% confidence interval using the same data?
3. How does the confidence interval change if the standard deviation was calculated from a larger sample?
4. Can you determine the required sample size to achieve a margin of error of 2 points with 90% confidence?
5. What is the effect of increasing the confidence level to 95%?

Tip: In hypothesis testing, a smaller sample size generally results in a wider confidence interval.