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. 100 170 133 93 75 94 116 100 85

Answer 1

(a) The sample mean startup cost is 107.3 thousand dollars.

The sample mean startup cost is 107.3 thousand dollars.

(b) The 90% confidence interval for the population average startup costs for candy store franchises is ($89.4, $125.2) thousand.

Explanation:

The questions asked related to the data are:

(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

(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

Solution:

The data provided is:

X = {100, 170, 133, 93, 75, 94, 116, 100, 85}

(a)

Compute the sample mean as follows:

`\bar x=(1)/(n)\sum X_(i)`

`=(1)/(9)* (100+170+133+93+75+94+116+100+85)`

`=(966)/(9)`

`=107.3`

The sample mean startup cost is 107.3 thousand dollars.

Compute the sample standard deviation as follows:

`s=\sqrt{(1)/(n-1)\sum (X_(i)-\bar x)^(2)}`

`=\sqrt{(1)/(9-1)* [(100-107.33)^(2)+(170-107.33)^(2)+...+(85-107.33)^(2)]}`

`= \sqrt{ ( 6696 )/( 9 - 1) } \\= 28.931\\\approx28.9`

The sample standard deviation of startup cost is 28.9 thousand dollars.

(b)

As the population standard deviation is not known we will use a t-interval.

The critical value of t for 90% confidence level and (n - 1) = 8 degrees of freedom is:

`t_(\alpha/2, (n-1))=t_(0.10/2, (9-1))=t_(0.05, 8)=1.86`

*Use a t-table for the value.

Compute the 90% confidence interval for population mean as follows:

`CI=\bar x\pm t_(\alpha/2, (n-1))* (s)/(√(n))`

`=107.3\pm 1.86* (28.9)/(√(9))`

`=107.3\pm 17.918\\=(89.382, 125.218)\\\approx (89.4, 125.2)`

Thus, the 90% confidence interval for the population average startup costs for candy store franchises is ($89.4, $125.2) thousand.