Question

How much does a sleeping bag cost? Let's say you want a sleeping bag that should keep you warm in temperatures from 20 degrees F to 45 degrees . A random sample of prices for sleeping bags in this temperature range was taken from Backpacker Magazine: Gear Guide (Vol.25 Issue 157, No 2) Brand names include American Camper, Cabela's Camp 7, Caribou, Cascade, and Coleman 80,90,100,120,75,37,30,23,100,110 105,95,105,60,110,120,95,90,60,70 A) Use a calculator with mean and sample standard deviation keys to verify that sample mean is around $83.75 and S is around $28.97 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. C) What does the confidence interval mean in the context of this problem?

Answer 1

a)
`\bar X =(\sum_(i=1)^n x_i)/(n)=(1675)/(20)=83.75`

`s=\sqrt{(\sum_(i=1)^n (x_i -\bar X)^2)/(n-1)}=28.97`

b) The 90% confidence interval would be given by (72.551;94.949)

c)We are 90% confident that the true mean temperature for the sleeping bags it's between 72.551 and 94.949

Explanation:

Data set given

80,90,100,120,75,37,30,23,100,110 105,95,105,60,110,120,95,90,60,70

Part a

We can calculate the sample mean and the sample deviation with the following formulas:

`\bar X =(\sum_(i=1)^n x_i)/(n)=(1675)/(20)=83.75`

`s=\sqrt{(\sum_(i=1)^n (x_i -\bar X)^2)/(n-1)}=28.97`

Part b

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=20-1=19`

Since the Confidence is 0.90 or 90%, the value of
`\alpha=0.1` and
`\alpha/2 =0.05`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,19)".And we see that
`t_(\alpha/2)=1.73`

Now we have everything in order to replace into formula (1):

`83.75-1.73(28.97)/(√(20))=72.551`

`83.75+1.73(28.97)/(√(20))=94.949`

So on this case the 90% confidence interval would be given by (72.551;94.949)

Part c

We are 90% confident that the true mean temperature for the sleeping bags it's between 72.551 and 94.949