SOLUTION: The prices (in dollars) for a graphing calculator are shown below for 8 online vendors. Estimate the population standard deviation in price with 90% confidence. 148 133 136 152

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Question 1086260: The prices (in dollars) for a graphing calculator are shown below for 8 online vendors. Estimate the population standard deviation in price with 90% confidence.
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Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Confidence Level = C = 90% = 0.90

alpha = 1-C = 1-0.90 = 0.10

With confidence intervals, we have a shaded region in the middle and two unshaded tails (see the image labeled "chi-square distribution" below)

The shaded region has area of 0.90 and the two unshaded regions combine to 0.10. The total area under the entire curve is 1.

Each tail has area of alpha/2 = 0.10/2 = 0.05

The sample size is n = 8, so the degrees of freedom are df = n-1 = 8-1 = 7

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Let's define

= left Chi-Square critical value

= right Chi-Square critical value

The goal is to find the critical values such that

where is the Chi-Square symbol

Again, see the image labeled "chi-square distribution" below.

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Use a calculator or table to find the Chi-Square critical values.

I'm going to use a stats table.

According to this table, it states that

You're probably wondering how I got these values. Well first you would highlight everything in the row that starts with 7 (to indicate df = 7). Then mark the columns 0.95 and 0.05 as I've done so in red

The two values at the row/column intersections are 2.167 and 14.067 in that order (from left to right)

These two values 2.167 and 14.067 are set up in such a way that

as shown by the diagram below

Point A = location of the left chi-square critical value
Point B = location of the right chi-square critical value
area under the curve between A and B = 0.90 = 90% = confidence level
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Now that we have the Chi-Square critical values, let's find the standard deviation. To make things simple, it's best to use a calculator to compute s. We can't find sigma (because we're trying to approximate it based on the sample) but we can find the sample estimate s.

Using a calculator, the approximate value of s is

It's possible to use other calculators such as the TI83. If you wish to use that, type the eight data values into a blank list (say L1) and then you can do a one-var stats on the data list. See this page for a similar example.

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Finally we can compute the confidence interval estimate for sigma

I'm going to use the formula drawn from this page (though I'm going to use slightly different notation for the chi-square symbols)

We are 90% confident that the population standard deviation (sigma = ) is between 5.359 and 13.654.