Question 1207795
**1. Calculate Sample Mean and Standard Deviation**

* **Sample Mean (x̄):** 
   (54.9 + 51.5 + 61.5 + 66.5 + 68.2 + 71.5 + 76.5 + 74.9 + 77.9 + 91.5 + 81.5) / 11 
   = 69.27

* **Sample Standard Deviation (s):** 
   Use a calculator or statistical software to calculate the sample standard deviation. 
   s ≈ 11.783

**2. Calculate the t-statistic**

* **Formula:**
   t = (x̄ - μ₀) / (s / √n) 
   where:
      * x̄ is the sample mean (69.27)
      * μ₀ is the hypothesized population mean (you did not provide this value - please specify the value of μ₀ for the accurate calculation)
      * s is the sample standard deviation (11.783)
      * n is the sample size (11)

* **Example:** 
   * Let's assume the hypothesized population mean (μ₀) is 60.
   * t = (69.27 - 60) / (11.783 / √11) 
   * t ≈ 2.618 

**3. Calculate the p-value**

* **Degrees of Freedom:** df = n - 1 = 11 - 1 = 10

* **Using a t-distribution table or statistical software:** 
   * Find the p-value associated with the calculated t-statistic (2.618) and degrees of freedom (10). 
   * **Note:** Since you did not specify the direction of the alternative hypothesis (Ha), we will assume a two-tailed test.

* **Example:** 
   * If t = 2.618 and df = 10, the p-value for a two-tailed test is approximately 0.0272.

**Therefore:**

* **test statistic = 2.618 (assuming μ₀ = 60)**
* **p-value = 0.0272 (assuming μ₀ = 60 and a two-tailed test)**

**Important Notes:**

* **Hypothesized Mean (μ₀):** You must specify the hypothesized population mean (μ₀) to accurately calculate the t-statistic and p-value. 
* **Software:** Use statistical software (like R, Python, Excel, or SPSS) to perform these calculations more efficiently and accurately.
* **Interpretation:** If the p-value is less than the significance level (α = 0.004), you would reject the null hypothesis. If the p-value is greater than or equal to α, you would fail to reject the null hypothesis.

This analysis provides a framework for conducting the t-test. Remember to adjust the calculations based on the specific hypothesized population mean (μ₀) for your analysis.