Respuesta :
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:
[tex]\bar x=\frac{1}{n}\sum X_{i}[/tex]
[tex]=\frac{1}{9}\times (100+170+133+93+75+94+116+100+85)[/tex]
[tex]=\frac{966}{9}[/tex]
[tex]=107.3[/tex]
The sample mean startup cost is 107.3 thousand dollars.
Compute the sample standard deviation as follows:
[tex]s=\sqrt{\frac{1}{n-1}\sum (X_{i}-\bar x)^{2}}[/tex]
[tex]=\sqrt{\frac{1}{9-1}\times [(100-107.33)^{2}+(170-107.33)^{2}+...+(85-107.33)^{2}]}[/tex]
[tex]= \sqrt{ \frac{ 6696 }{ 9 - 1} } \\= 28.931\\\approx28.9[/tex]
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:
[tex]t_{\alpha/2, (n-1)}=t_{0.10/2, (9-1)}=t_{0.05, 8}=1.86[/tex]
*Use a t-table for the value.
Compute the 90% confidence interval for population mean as follows:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
[tex]=107.3\pm 1.86\times \frac{28.9}{\sqrt{9}}[/tex]
[tex]=107.3\pm 17.918\\=(89.382, 125.218)\\\approx (89.4, 125.2)[/tex]
Thus, the 90% confidence interval for the population average startup costs for candy store franchises is ($89.4, $125.2) thousand.
