To calculate the t-score, we first need to understand what specific t-score you're looking for. However, I will assume you are referring to the t-score for a particular data point within a sample, which measures how far that point is from the sample mean in terms of standard deviations, adjusted by the sample size.

Steps to Calculate the T-Score:

1. Calculate the Mean (μ): $\text{Mean} = \frac{\sum X_i}{n}$ where $X_i$ represents each data point and $n$ is the number of data points.

2. Calculate the Sample Standard Deviation (s): $s = \sqrt{\frac{\sum (X_i - \mu)^2}{n-1}}$

3. Calculate the T-Score for Each Data Point: $t = \frac{X_i - \mu}{\frac{s}{\sqrt{n}}}$ Here, $X_i$ is the data point, $\mu$ is the mean, $s$ is the standard deviation, and $n$ is the sample size.

Let's proceed with the calculations.

1. Calculate the Mean (μ)

$\mu = \frac{80 + 75 + 60 + 32 + 48 + 54 + 72 + 64 + 84 + 98 + 78 + 40}{12}$

2. Calculate the Sample Standard Deviation (s)

$s = \sqrt{\frac{(X_i - \mu)^2}{11}}$

3. Calculate the T-Score for each data point.

I will calculate these now.### Results:

1. Mean (μ): $65.42$ (rounded to two decimal places)
2. Sample Standard Deviation (s): $19.40$ (rounded to two decimal places)
3. T-Scores for Each Data Point:
   • For 80: $2.60$
   • For 75: $1.71$
   • For 60: $-0.97$
   • For 32: $-5.97$
   • For 48: $-3.11$
   • For 54: $-2.04$
   • For 72: $1.18$
   • For 64: $-0.25$
   • For 84: $3.32$
   • For 98: $5.82$
   • For 78: $2.25$
   • For 40: $-4.54$

These t-scores indicate how many standard deviations each data point is from the sample mean, adjusted for the sample size.

Do you want more details or have any questions?

Related Questions:

1. What is the significance of a high or low t-score in this context?
2. How do t-scores relate to confidence intervals?
3. Can we use t-scores to determine if a particular data point is an outlier?
4. What assumptions are made when using t-scores?
5. How does sample size affect the t-score?
6. How would the t-scores change if we added more data points to the sample?
7. What is the difference between a t-score and a z-score?
8. How can t-scores be used in hypothesis testing?

Tip: T-scores are particularly useful when dealing with smaller sample sizes or when the population standard deviation is unknown.