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

Answer:

(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.

Step-by-step 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:

$\bar x=\frac{1}{n}\sum X_{i}$

  $=\frac{1}{9}\times (100+170+133+93+75+94+116+100+85)$

  $=\frac{966}{9}$

  $=107.3$

The sample mean startup cost is 107.3 thousand dollars.

Compute the sample standard deviation as follows:

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

  $=\sqrt{\frac{1}{9-1}\times [(100-107.33)^{2}+(170-107.33)^{2}+...+(85-107.33)^{2}]}$

  $= \sqrt{ \frac{ 6696 }{ 9 - 1} } \\= 28.931\\\approx28.9$

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:

$CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}$

     $=107.3\pm 1.86\times \frac{28.9}{\sqrt{9}}$

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

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