Question 1185508: Do you want to own your own candy store? Wow! With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below. Assume that the population of x values has an approximately normal distribution.
96 178 126 94 75 94 116 100 85
A button hyperlink to the SALT program that reads: Use SALT.
(a) Use a calculator with mean and sample standard deviation keys to find the sample mean startup cost x and sample standard deviation s. (Round your answers to four decimal places.)
x =
thousand dollars
s =
thousand dollars
(b) Find a 90% confidence interval for the population average startup costs 𝜇 for candy store franchises. (Round your answers to one decimal place.)
lower limit
thousand dollars
upper limit
thousand dollars
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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