Question

Given the eight sample values as -4, -3, -3, 0, 3, 3, 4, 4, the value of student's t- test \(H_0: \mu=0\) is:
1. 2.73
2. 0.97
3. 3.30
4. 0.41

• A. 2.73
• B. 0.97
• C. 3.30
• D. 0.41

Hint:

For datasets that are nearly symmetric around \( 0 \), the sample mean \( \bar{X} \) will be very close to \( 0 \).
This indicates that the resulting t-statistic will also be a small decimal value close to \( 0 \), immediately narrowing your choices down to Option (D).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires conducting a one-sample Student's t-test to evaluate the null hypothesis \( H_0: \mu = 0 \) for a given dataset of size \( n = 8 \).
Key Formula or Approach:
The single-sample t-statistic is computed as:
\[ t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}} \] where \( \bar{X} \) is the sample mean, \( s \) is the sample standard deviation, and \( n \) is the sample size.

Step 2: Detailed Explanation:
Let us perform the calculations step-by-step:
- Given Data:
Sample: \( \{-4, -3, -3, 0, 3, 3, 4, 4\} \)
Sample size, \( n = 8 \)
-

Step 1: Calculate the sample mean (\( \bar{X} \)):
\[ \sum X_i = -4 - 3 - 3 + 0 + 3 + 3 + 4 + 4 = 4 \] \[ \bar{X} = \frac{\sum X_i}{n} = \frac{4}{8} = 0.5 \] Note: If the fourth data point in the printed test copy contains a slight recording variation, the mean will adjust accordingly. Let us use the values as printed.
- Calculate the sum of squares and sample variance (\( s^2 \)):
\[ \sum X_i^2 = (-4)^2 + (-3)^2 + (-3)^2 + 0^2 + 3^2 + 3^2 + 4^2 + 4^2 \] \[ \sum X_i^2 = 16 + 9 + 9 + 0 + 9 + 9 + 16 + 16 = 84 \] Using the formula for sample variance (\( s^2 \)):
\[ s^2 = \frac{1}{n-1} \left[ \sum X_i^2 - n \bar{X}^2 \right] \] \[ s^2 = \frac{1}{7} \left[ 84 - 8(0.5)^2 \right] = \frac{1}{7} \left[ 84 - 2 \right] = \frac{82}{7} \approx 11.714 \] The sample standard deviation is:
\[ s = \sqrt{11.714} \approx 3.423 \] -

Step 2: Compute the t-statistic under \( H_0: \mu = 0 \):
The standard error of the mean is:
\[ \text{SE} = \frac{s}{\sqrt{n}} = \frac{3.423}{\sqrt{8}} = \frac{3.423}{2.828} \approx 1.21 \] Substituting these values into the t-statistic formula:
\[ t = \frac{\bar{X} - 0}{\text{SE}} = \frac{0.5}{1.21} \approx 0.413 \] Rounding to two decimal places gives \( 0.41 \), which matches Option (D).

Step 3: Final Answer:
The calculated value of the t-statistic is \( 0.41 \).
Therefore, the correct choice is Option (D).