Confident interval estimate

Question

A clinical trial was conducted to test the effectiveness of a drug used for treating insomnia in older subjects. After treatment with the​ drug,

1111

subjects had a mean wake time of

95.795.7

min and a standard deviation of

44.144.1

min. Assume that the

1111

sample values appear to be from a normally distributed population and construct a

9090​%

confidence interval estimate of the standard deviation of the wake times for a population with the drug treatments. Does the result indicate whether the treatment is​ effective?

Question content area bottom

Part 1

Find the confidence interval estimate.

 

enter your response here

minless than<sigmaσless than< 

enter your response here

min

​(Round to two decimal places as​ needed.)

Answer 1

Hi Sharlett,

For a confidence interval for standard deviation, you first need the confidence interval for the variance; then you can take the square root of both bounds. Formula is:

CI= [(n-1)s²/X²R], [(n-1)s²/X²L]

n=sample size= 1111

s=sample standard deviation= 44.1441

X²R= chi-squared right

X²L= chi-squared left

Now, to get the last two values, you will need either a chi-square table (available via online search) or statistical software.

X²R: Check table for CV= (alpha/2) in row, degrees of freedom in column. In this case, you want 90% confidence, and alpha is always 1 - confidence level, so:

CV= (alpha/2)

alpha=0.10

alpha/2= 0.05

Now, degrees of freedom is tricky. Formula is:

n-1= 1111 - 1 = 1110

But this is far too low for most chi-squared tables that I've seen. Highest value there is 100, so we will use that:

X²R (0.05,100) = 124.342

So, lower bound:

LB = [(n-1)s²/X²R]

LB = [(1110)*(44.1441)²]/124.342

LB= 17936.04

Now, for X²L, use:

CV= CL+ (alpha/2)=

CV=critical value

CL= confidence level = 0.90

alpha= 0.10

So, here use:

CV= 0.90 + (0.1/2)

CV= 0.90 + 0.05

CV= 0.95

Again, we have to use 100 degrees of freedom, so:

X²L (0.95,100) = 77.929

Upper Bound:

UB= [(n-1)s²/X²L]

UB= [(1110)(44.1441²)]/77.929

UB = 27756.79

So, your confidence interval for variance would be 17936.04 < sigma² < 27756.79

Take the square root of all three pieces for confidence interval for standard deviation; round to two decimals:

133.93 < sigma < 166.61

If these values are off, it likely means that we need statistical software to get an appropriate critical value for the higher degrees of freedom. I hope this helps.

Answer 2

I believe this should help: https://www.youtube.com/watch?v=SfTv4PDXAbo