A certain test preparation course is designed to help students improve their scores on the MCAT exam.  A mock exam is given at the beginning and end of the course to determine the effectiveness of the course.  The following measurements are the net change in 3
 students' scores on the exam after completing the course:

1,6,5

Using these data, construct a 98%
 confidence interval for the average net change in a student's score after completing the course.  Assume the population is approximately normal.


Step 2 of 4 :  Calculate the sample standard deviation for the given sample data. Round your answer to one decimal place.'

To calculate the sample standard deviation for the given data, we follow these steps:

### Step 1: Find the sample mean.
The data points are: \(1, 6, 5\).

\[
\text{Mean} (\bar{x}) = \frac{1 + 6 + 5}{3} = \frac{12}{3} = 4
\]

### Step 2: Calculate the squared differences from the mean.
Now, we subtract the mean from each data point and square the result:

\[
(1 - 4)^2 = (-3)^2 = 9
\]
\[
(6 - 4)^2 = (2)^2 = 4
\]
\[
(5 - 4)^2 = (1)^2 = 1
\]

### Step 3: Find the sum of squared differences.
Add the squared differences:

\[
9 + 4 + 1 = 14
\]

### Step 4: Divide by \(n-1\) to get the variance.
Since this is a sample, we divide by \(n-1 = 3-1 = 2\):

\[
\text{Variance} = \frac{14}{2} = 7
\]

### Step 5: Take the square root to get the sample standard deviation.

\[
\text{Standard deviation} = \sqrt{7} \approx 2.6
\]

So, the sample standard deviation is **2.6**.

Would you like details on how to proceed to the next steps or any clarifications?

Here are five related questions:
1. How is the sample standard deviation different from population standard deviation?
2. Why do we divide by \(n-1\) for sample variance instead of \(n\)?
3. What assumptions are necessary for constructing a confidence interval?
4. How does increasing the confidence level affect the interval width?
5. What are the implications of having a small sample size on the confidence interval?

**Tip**: Always round your final answer according to the problem’s instructions, but keep more decimal places during intermediate steps for accuracy.

A certain test preparation course is designed to help students improve their scores on the MCAT exam. A mock exam is given at the beginning and end of the course to determine the effectiveness of the course. The following measurements are the net change in 3 students' scores on the exam after completing the course: 1, 6, 5. Using these data, construct a 98% confidence interval for the average net change in a student's score after completing the course. Assume the population is approximately normal. Step 2 of 4: Calculate the sample standard deviation for the given sample data. Round your answer to one decimal place.

Learn how to calculate the sample standard deviation for MCAT score improvements using the data set {1, 6, 5}. Follow the step-by-step guide to find the sample variance and standard deviation, with a detailed explanation of confidence intervals and statistical concepts. Ideal for undergraduate-level statistics students.

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