Question

The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1194

1306

1264

1180

1268

1316

1275

1317

1275

(a) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit	A.D.
upper limit	A.D.

How much does a sleeping bag cost? Let's say you want a sleeping bag that should keep you warm in temperatures from 20°F to 45°F. A random sample of prices ($) for sleeping bags in this temperature range is given below. Assume that the population of x values has an approximately normal distribution.

80	35	55	75	50	90	30	23	100	110
105	95	105	60	110	120	95	90	60	70

(b) Using the given data as representative of the population of prices of all summer sleeping bags, find a 90% confidence interval for the mean price μ of all summer sleeping bags. (Round your answers to two decimal places.)

lower limit	$
upper limit	$

How much do wild mountain lions weigh? Adult wild mountain lions (18 months or older) captured and released for the first time in the San Andres Mountains gave the following weights (pounds):

69

110

128

130

60

64

(c) Find a 75% confidence interval for the population average weight μ of all adult mountain lions in the specified region. (Round your answers to one decimal place.)

lower limit	lb
upper limit	lb

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

178

134

94

75

94

116

100

85

(d) 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 1

a)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   49.3088
Sample Size ,   n =    9
Sample Mean,    x̅ = ΣX/n =    1266.1111

Level of Significance ,    α =    0.1          
degree of freedom=   DF=n-1=   8          
't value='   tα/2=   1.8595   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   49.309   / √   9   =   16.4363
margin of error , E=t*SE =   1.8595   *   16.436   =   30.564
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    1266.11   -   30.564   =   1235.5471
Interval Upper Limit = x̅ + E =    1266.11   -   30.564   =   1296.6752

lower limit	1236 A.D.
upper limit	1297 A.D.

==================================

b)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   28.8698
Sample Size ,   n =    20
Sample Mean,    x̅ = ΣX/n =    77.9000

Level of Significance ,    α =    0.1          
degree of freedom=   DF=n-1=   19          
't value='   tα/2=   1.7291   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   28.870   / √   20   =   6.4555
margin of error , E=t*SE =   1.7291   *   6.455   =   11.162
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    77.90   -   11.162   =   66.74
Interval Upper Limit = x̅ + E =    77.90   -   11.162   =   89.06

=============================

c)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   32.8253
Sample Size ,   n =    6
Sample Mean,    x̅ = ΣX/n =    93.5000

Level of Significance ,    α =    0.25          
degree of freedom=   DF=n-1=   5          
't value='   tα/2=   1.3009   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   32.825   / √   6   =   13.4009
margin of error , E=t*SE =   1.3009   *   13.401   =   17.434
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    93.50   -   17.434   =   76.1
Interval Upper Limit = x̅ + E =    93.50   -   17.434   =   110.9

d)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   31.1774
Sample Size ,   n =    9
Sample Mean,    x̅ = ΣX/n =    108.4444

Level of Significance ,    α =    0.1          
degree of freedom=   DF=n-1=   8          
't value='   tα/2=   1.8595   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   31.177   / √   9   =   10.3925
margin of error , E=t*SE =   1.8595   *   10.392   =   19.325
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    108.44   -   19.325   =   89.1
Interval Upper Limit = x̅ + E =    108.44   -   19.325   =   127.8