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Step-By-Step Example of a Confidence Interval for a Mean, sigma unknown (used Ex 8.7)
Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for 15 random subjects with the results given below.
8.6, 9.4, 7.9, 6.8, 8.3, 7.3, 9.2, 9.6, 8.7, 11.4, 10.3, 5.4, 8.1, 5.5, 6.9
Use the sample data to construct a 95% confidence interval for the mean sensory rate for the populations (assumed normal) from which you took this data.
| Guidelines | Example |
|---|---|
| Plan: State what we need to know. | We are asked to find a 95% confidence interval for the mean sensory rate, μ, of acupuncture subjects. We have a sample of 15 rates. We do not know the population sigma. |
| Model: Think about the assumptions and check the conditions. |
Randomization Condition: The sample is a random sample.
Even though the data is slightly skewed, it is unimodal, and there are no outliers, so we can use the model. |
| State the parameters and the sampling model | The conditions are satisfied and σ is unknown, so we will use a confidence interval for a mean with unknown standard deviation.We need the sample mean and Margin of Error (ME).
= 8.2267; s = 1.6722; n =15;
df = 15-1 = 14; ME =
|
Mechanics: CL = 0.95, so α = 1-CL = 1-0.95 = 0.05.
= 0.25;
= t
0.025
The area to the right of t 0.25 is 1 - 0.025 = 0.975, so t 0.025, 14 = 2.14 |
ME =2.14
= - ME = 8.2267 - 0.9240 = 7.3027
= + ME = 8.2267 + 0.9240 = 9.1507
|
| Conclusion: Interpret your result in the proper context, and relate it to the original question. | I am 95% confident that the interval from 7.30 to 9.15 contains the true mean score of all the sensory rates. 95% of all confidence intervals constructed in this way contain the true mean sensory rate score. |
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