Download Activity #34

Math 103 - Cooley
Statistics for Teachers
OCC
Activity #34 – Confidence Intervals
California State Content Standard - Statistics, Data Analysis, and Probability
N/A
Sampling Distribution of the Sample Mean
Suppose that a variable x of a population has mean μ and standard deviation σ. Then, for samples of size n,
 The mean of variable x equals the population mean, or  x   ;
 The standard deviation of x equals the population standard deviation divided by the square root of
the sample size, or

.
n
 If x is normally distributed, so is x , regardless of sample size; and
 If the sample size is large, x is approximately normally distributed, regardless of the distribution
of x.
Confidence Interval For The Mean Of A Normal Population
Draw an SRS of size n from a Normal population having unknown mean µ and known standard deviation
σ. A level (1 – α) confidence interval for µ is
x  z
The critical value z
2

2
n
is illustrated below and found in Table II.
The One-Mean z-Interval Procedure For A Population Mean
Assumptions
1) Simple random sample
2) Normal population or large sample (n ≥ 30)
3) σ known
Step 1 – For a confidence level (1 – α), use Table I to find z 2 .
Step 2 – The confidence interval for µ is from
x  z
where z
2

2
to x  z

2
n
n
is found in Step 1, n is the sample size, and x is computed from the sample data.
Step 3 – Interpret the confidence interval.
The confidence interval is exact for normal populations and is approximately correct for large samples
from nonnormal populations.
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 Exercises:
1)
You measure the lifetime of a random sample of 25 tires of a certain brand. The sample mean is
x = 50 months. Suppose that the lifetimes for tires of this brand follow a normal distribution, with
unknown mean µ and standard deviation σ = 5 months. Find a 95% confidence interval for µ.
2)
A medical researcher treats 145 subjects with high cholesterol with a new drug. The average
decrease in cholesterol level is x = 80 after two months of taking the drug. Assume that the
decrease in cholesterol after two months of taking the drug follows a normal distribution, with
unknown mean µ and standard deviation σ = 20. Find a 95% confidence interval for µ. Then find a
99% confidence interval for µ.
3)
The time (in number of days until maturity of a certain variety of tomato plant is normally
distributed with mean µ and standard deviation σ = 2.4. If you select a simple random sample of
four plants of this variety and measure the time until maturity. The four times, in days are: 63, 69,
62, 66. Based on these data, find a 90% confidence interval for µ.
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Math 103 - Cooley
Statistics for Teachers
OCC
Activity #34 – Confidence Intervals
Interpretation and the Meaning of Confidence Intervals for µ
Consider example #1:
You measure the lifetime of a random sample of 25 tires of a certain brand. The sample
mean is x = 50 months. Suppose that the lifetimes for tires of this brand follow a normal
distribution, with unknown mean µ and standard deviation σ = 5 kg. Find a 95%
confidence interval for µ.
During class we found that the 95% confidence interval for the population mean, µ, was (48.04, 51.96).
Students and teachers sometimes refer to this by saying that “We are 95% confident that µ lies in the
interval (48.04, 51.96)”.
However, while this wording is okay for student use in the classroom, it is technically incorrect, and would
not be used by actual statisticians when interpreting the meaning of a confidence interval. Because you
would never describe yourself as being 75% happy or 50% in love with someone, you also can’t be 95%
confident about something.
In actuality, a 95% confidence interval means that the method used to obtain the confidence interval
produces an interval that encapsulates the unknown, µ, 95% of the time. To be more precise a
statistician would say that if we were to take 100 samples of size 25 from the population, and from those
samples construct 100 sample means, and then construct 100 confidence intervals around the sample
means at the 95% level, then 95 of the 100 confidence intervals constructed from the samples would
include the population mean, while the other 5 of the 100 would not. So, for an individual sample of size
25 from the tire population there is a 95% chance that its corresponding confidence interval encapsulates
the population mean.
When interpreting the wording of confidence intervals for µ taken at level 1 – α taken from a sample of
size n, the following phrases are equivalent:


 
 .
 We are (1 – α) % confident that µ lies in the interval  x  z 2
, x  z 2
n
n

 (1 – α) % of all samples of size n taken produce an interval that contains the population mean µ.
 The method used to obtain the confidence interval produces an interval that encapsulates the
unknown, µ, (1 – α) % of the time.
 In 100 samples of size n taken from the population, (1 – α) of then will produce confidence
intervals that contain the population mean µ.
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