Question

In: Statistics and Probability

How much does a sleeping bag cost? Let's say you want a sleeping bag that should...

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.

45 65 105 110 120 55 30 23 100 110
105 95 105 60 110 120 95 90 60 70

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean price x and sample standard deviation s. (Round your answers to two decimal places.)

x = $  
s = $  


(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     $  

Solutions

Expert Solution

Values ( X ) Σ ( Xi- X̅ )2
45 1493.8225
65 347.8225
105 455.8225
110 694.3225
120 1321.3225
55 820.8225
30 2878.3225
23 3678.4225
100 267.3225
110 694.3225
105 455.8225
95 128.8225
105.0 455.8225
60 559.3225
110 694.3225
120 1321.3225
95 128.8225
90 40.3225
60 559.3225
70 186.3225
Total 1673 17182.55

Part a)

Mean X̅ = Σ Xi / n
X̅ = 1673 / 20 = 83.65
Sample Standard deviation SX = √ ( (Xi - X̅ )2 / n - 1 )
SX = √ ( 17182.55 / 20 -1 ) = 30.07

Part b)

Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
t(α/2, n-1) = t(0.1 /2, 20- 1 ) = 1.729
83.65 ± t(0.1/2, 20 -1) * 30.0723/√(20)
Lower Limit = 83.65 - t(0.1/2, 20 -1) 30.0723/√(20)
Lower Limit = 72.02
Upper Limit = 83.65 + t(0.1/2, 20 -1) 30.0723/√(20)
Upper Limit = 95.28
90% Confidence interval is ( 72.02 , 95.28 )



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