80,75,60,32,48,54,72,64,84,98,78,40 calculate the t score

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.

Learn how to calculate the t-score for a given data set step-by-step. This problem covers statistical concepts including sample mean, standard deviation, and t-score calculations for each data point. Suitable for advanced high school or college-level students.

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