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.

Question

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.

94 177 127 94 75 94 116 100 85
(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 one decimal place.)
x = thousand dollars
s = thousand dollars

Answer 1

I don't have a calculator with those keys, but here is what you can do.

Find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

I'll let you do the calculations.

Answer 2

To find the sample mean startup cost (x) and sample standard deviation (s), you need to compute the mean and standard deviation of the given random sample of startup costs.

To do this, follow these steps:

Step 1: Add up all the values in the sample (startup costs).
94 + 177 + 127 + 94 + 75 + 94 + 116 + 100 + 85 = 962

Step 2: Count the number of values in the sample.
In this case, there are 9 values.

Step 3: Calculate the sample mean (x) by dividing the sum of the values by the number of values.
x = Sum of values / Number of values = 962 / 9 = 106.9 thousand dollars (rounded to one decimal place)

Step 4: Calculate the sample standard deviation (s) using the following formula:

s = sqrt((sum of squared differences) / (number of values - 1))

First, calculate the squared difference for each value by subtracting the mean from each value, squaring the result, and then summing them up.

Squared Differences:
(94 - 106.9)^2 = 150.61
(177 - 106.9)^2 = 4593.61
(127 - 106.9)^2 = 409.61
(94 - 106.9)^2 = 150.61
(75 - 106.9)^2 = 1015.21
(94 - 106.9)^2 = 150.61
(116 - 106.9)^2 = 81.00
(100 - 106.9)^2 = 47.61
(85 - 106.9)^2 = 481.61

Sum of Squared Differences: 150.61 + 4593.61 + 409.61 + 150.61 + 1015.21 + 150.61 + 81.00 + 47.61 + 481.61 = 7000.88

Next, divide the sum of squared differences by the number of values minus 1 and take the square root of the result.

s = sqrt(7000.88 / (9 - 1)) = sqrt(7000.88 / 8) ≈ 29.3 thousand dollars (rounded to one decimal place)

Therefore, the sample mean startup cost (x) is approximately 106.9 thousand dollars, and the sample standard deviation (s) is approximately 29.3 thousand dollars.