Question 1185508
Here's how to calculate the sample mean, sample standard deviation, and the 90% confidence interval:

**(a) Sample Mean and Standard Deviation:**

1. **Sample Mean (x̄):**  Sum all the startup costs and divide by the number of stores (n = 9):

   x̄ = (96 + 178 + 126 + 94 + 75 + 94 + 116 + 100 + 85) / 9
   x̄ = 964 / 9
   x̄ ≈ 107.1111 thousand dollars

2. **Sample Standard Deviation (s):**

   First, calculate the squared differences from the mean for each value, sum them, divide by (n-1), and then take the square root:

   s = √[Σ(xi - x̄)² / (n - 1)]

   Here's a breakdown:

   * (96-107.11)² = 123.46
   * (178-107.11)² = 4997.74
   * (126-107.11)² = 356.66
   * (94-107.11)² = 171.86
   * (75-107.11)² = 1031.54
   * (94-107.11)² = 171.86
   * (116-107.11)² = 78.94
   * (100-107.11)² = 50.54
   * (85-107.11)² = 488.94

   Sum of squared differences: 7471.22
   s = √(7471.22 / 8)
   s ≈ √933.9025
   s ≈ 30.56 thousand dollars

**(b) 90% Confidence Interval:**

Since the population standard deviation is unknown and the sample size is small (n < 30), we use a t-distribution.

1. **Degrees of Freedom (df):** df = n - 1 = 9 - 1 = 8

2. **Critical Value (t_c):** For a 90% confidence level and df = 8, look up the t-value in a t-table or use a calculator. The t_c ≈ 1.860.

3. **Margin of Error (E):**

   E = t_c * (s / √n)
   E = 1.860 * (30.56 / √9)
   E = 1.860 * (30.56 / 3)
   E ≈ 19.01

4. **Confidence Interval:**

   Lower Limit = x̄ - E = 107.11 - 19.01 ≈ 88.1 thousand dollars
   Upper Limit = x̄ + E = 107.11 + 19.01 ≈ 126.1 thousand dollars

Therefore:

* x̄ = 107.1111 thousand dollars
* s = 30.5600 thousand dollars
* Lower Limit = 88.1 thousand dollars
* Upper Limit = 126.1 thousand dollars