In: Statistics and Probability
Estimate Confidence Interval for Population Mean based on sample data and using Appendix Table for t-Distribution
(Population standard deviation is not known, but sample standard deviation is given).
Read material from our e-textbook (link eText) pages 314-330.
Here is an example with steps you can follow: sample size n=9, sample mean=80,
sample standard deviation s=25
(population standard deviation is not known)
Estimate confidence interval for population mean with confidence
level 90%.
Confidence Interval = Sample Mean ± Margin of Error Margin of Error = (t-value)×s/?n (see page 328)
t-value should be taken from Appendix Table IV.
For n=9 df=n-1=9-1=8
For Confidence Level 90% a = 1 - 0.90 = 0.10, a/2 = 0.10/2 =
0.05
So, we are looking for t-value in column t0.05 and row 8. t-value =
1.860
Margin of Error E = 1.860 * 25/?(9) = 15.5 ? 16 Confidence interval for population mean: 80 ± 16 or between 64 and 96.
Here is your DB-5 assignment: |
For sample mean x , choose a value between 10 and
80 |
1. Use Appendix table IV and find t-value for your case; 2.
Calculate Margin of Error; |
Solution:
Here, we have to find the 90% confidence interval for the population mean.
C = 90% = 0.90, ? = 1 – C = 1 – 0.90 = 0.10
We are not given a population standard deviation.
So, we have to use t confidence interval.
For sample size, we have to choose a value between 4 and 10.
Suppose, we select sample size = n = 9.
For sample mean Xbar, we have to choose a value between 10 and 80.
Suppose, we select sample mean = Xbar = 50
For sample standard deviation S, we have to choose a value between 3 and 20.
Suppose, we select sample standard deviation = S = 12.
Now, we have
Xbar = 50, S = 12, n = 9
C = 90% = 0.90, ? = 1 – C = 1 – 0.90 = 0.10, ?/2 = 0.10/2 = 0.05
df = n – 1 = 9 – 1 = 8
t-value = 1.8595
(By using t-table)
Confidence Interval = Sample Mean ± Margin of Error
Margin of error = t-value * S/sqrt(n)
Margin of error = 1.8595*12/sqrt(9)
Margin of error =1.8595*12/3
Margin of error =1.8595*4
Margin of error = 7.438192
Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval = 50 ± 7.438192
Lower limit = 50 - 7.438192 = 42.561808
Upper limit = 50 + 7.438192 = 57.438192
Confidence interval = (42.561808, 57.438192)
We are 90% confident that the population mean will be lies between 42.561808 and 57.438192.