# Random Variable and the Population Proportion Questions

Section 8.3 One Sample Interval MeanOutline for A Single Population Mean

Suppose you want to estimate the mean. This is the procedure for computing a confidence interval for the mean. You will see that this

is the same process we covered when discussing the corresponding hypothesis test.

Con dence Interval for One Population Mean (t-Interval)

Step 1. State the Random Variable and the Parameter in Words.

We write:

x =

μ =

random variable; and

mean of random variable.

Step 2. State and Check the Assumptions for a Hypothesis Test

1. A random sample of size n is taken.

2. The population of the random variable is normally distributed, though the t-test is fairly robust to the assumption if the sample size

is large. This means that if this assumption isn’t met, but your sample size is quite large (over 30), then the results of the t-test are

valid. If sample is small, we need to verify that the given data set is approximately normal.

Step 3. Find the Sample Statistic and Con dence Interval

The confidence interval is found by

x̄ − E < μ < x̄ + E
where
s
E = tc
−
−
√n
and
is the point estimator for μ;
t c is the critical value where C is the confidence level you are trying to estimate with and degrees of freedom: df
s is the sample standard deviation; and
n is the sample size.
x̄
= n − 1
;
Note: If you are doing a hypothesis test at a particular level of significance, α, then you can determine what confidence level to
estimate with if you then want to do a confidence interval. If your hypothesis test was one tailed, then C = 1 − 2α. If your hypothesis
test was two-tailed, then C = 1 − α.
Step 4. Statistical Interpretation
In general this looks like, “there is a C% chance that the statement contains the true mean.”
Step 5. Real World Interpretation
This is where you state what values the mean is between. You do not include the percentage, since the percentage is the chance the
interval is correct and not the change that the mean is in the interval.
The critical value is a value from the Student’s t-distribution. The critical values are found in table A.2 in the appendix
How to Check the Assumptions of Con dence Interval
In order for the confidence interval to be valid, the assumptions of the test must be true. Whenever you run a confidence interval, you
must make sure the assumptions are true. You need to check them. Here is how you do this:
1. For the assumption that the sample is a random sample, describe how you took the sample. Make sure your sampling technique is
random.
2. For the assumption that population is normal, remember the process of assessing normality from Chapter 6.
Example 8.3.1
The life expectancy for a person born in the year 2011 was estimated for randomly selected countries in Europe. Find a 95%
confidence interval for the mean life expectancy of people in Europe who were born in the year 2011. The data is
74, 75, 73, 71, 74, 69, 71, 77, 66, 76, 78, 74, 76, 81, 71, 81, 84, 78, 82, 85, 78
Solution.
We follow the step we have outlined.
Step 1. State the Random Variable and Parameter
Let x
μ =
=
life expectancy for a person born in 2011 in Europe
mean life expectancy for a person born in 2011 in Europe
Step 2. State and Check the Assumptions
1. A simple random sample of life expectancy for a person born in 2011 in Europe from 21 countries. The problem states that there is
a random sample taken, so the sample is a simple random sample.
2. The population of the life expectancy for all people born in 2011 in Europe is normally distributed. It is a good idea to check this out
anyways, in case you have forgotten.
Load the data set by
life

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