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Section 6-1 – Confidence Intervals for
the Mean (Large Samples)
• Estimating Population Parameters
VOCABULARY:
Point Estimate –
a single value estimate for a population parameter.
The most unbiased point estimate of the population
parameter is the sample mean.
Interval Estimate An interval, or range of values, used to estimate a
population parameter.
Level of Confidence - Denoted as c, it is the probability that the interval estimate
contains the population parameter.
Margin of Error Sometimes also called the maximum error of estimate, or
error tolerance. It is denoted as E, and is the greatest possible
distance between the point estimate and the value of the
parameter it is estimating.
c-confidence interval -Is found by adding and subtracting E from the sample mean.
The probability that the confidence interval contains µ is c.
FORMULAS:
Margin of Error =
𝑬 = 𝒛𝒄 𝝈𝒙 = 𝒛𝒄
σ
𝒏
To use the formula, it is assumed that the population standard deviation is
known.
This is rarely the case, but when 𝑛 ≥ 30, the sample standard deviation s can
be used in place of σ.
s
The formula effectively becomes 𝑬 = 𝒛𝒄
𝒛𝒄 = InvNorm of
1−𝑐
1
, or (1 −
2
2
c-confidence interval sample standard deviation:
𝒏
𝑐) (explained why on bottom of page 311)
𝒙−𝑬<𝝁<𝒙+𝑬
𝒔=
(𝒙−𝑥)𝟐
𝒏−𝟏
GUIDELINES:
1)Find the sample statistics 𝑛 and 𝑥.
𝑛 is the sample size.
𝑥 is the sample mean.
2)Specify σ, if known. Otherwise, if 𝑛 ≥ 30, find the sample standard deviation s
and use it as a point estimate for σ.
3)Find the critical value 𝑧𝑐 that corresponds to the given level of confidence.
The three most commonly used confidence levels are 90%, 95%, and 99%.
The corresponding 𝑧𝑐 values are:
90% -- 1.645
95% -- 1.96
99% -- 2.576
It would be beneficial to memorize these. You will be using them a lot.
GUIDELINES:
4)Find the margin of error E. (𝑬 = 𝒛𝒄
s
)
𝒏
5)Find the left and right endpoints and form the confidence interval.
𝒙−𝑬<𝝁<𝒙+𝑬
Now the good news!!
The TI-84 can help with this, too.
We are going to walk through Example 4 on page 314, using the data points from
Example 1 on page 310.
1)Enter the 50 data points into L1 on your calculator (STAT Edit).
2)STAT Calc 1 to find the sample standard deviation (we can use this because
we have 50 data points; 𝑛 ≥ 30).
s = 5.01
3)STAT TESTS 7 (Z-Interval)
4)Select Data, since you have the data entered into the calculator.
5)Enter 5.01 as the standard deviation, and .99 as the C-Level (level of
confidence).
5)Select Calculate to get the interval.
We can be 99% sure that the actual population mean is between 10.575
and 14.225.
Notice that the calculator also tells us that the mean of the data we
entered is 12.4, that the standard deviation of the data was 5.01 and that n
was 50.
If we need to know what E is, simply find the distance between the
interval endpoints and divide by 2.
(14.225 – 10.575)/2 = 1.825.
Look at Example 5 on page 315.
n = 20, 𝑥 = 22.9, σ = 1.5, and c = 90%
STAT TESTS 7, select Stats (since you are going to provide the stats instead of
the actual data points).
Enter 1.5, 22.9, 20, and .9 and then calculate.
We can be 90% certain that the actual mean age of all students currently
enrolled in college is between 22.3 and 23.5.
As a general rule, we round our interval endpoints to the same number
of decimal points as the mean that is given to us.
We were given 22.9, which is one decimal, so we rounded our interval
to one decimal place.
CALCULATING MINIMUM SAMPLE SIZE
How do you know how many experiments or trials are needed in order to
achieve the desired level of confidence for a given margin of error?
We take the formula for finding E and solve it for n.
s
𝒛 σ
𝐸 = 𝒛𝒄 becomes 𝑛 = ( 𝒄 )2 .
𝒏
𝐸
Remember, if you don’t know what σ is, you can use s, as long as 𝑛 ≥ 30.
Example 6 on page 316c = 95%, 𝑧𝑐 = 1.96, 𝜎 ≈ 𝑠 ≈ 5.01 (from Example 1), E = 1 (given).
𝑛=(
𝒛𝒄 σ 2
1.96(5.0) 2
9.8
) =(
) = ( )2 =
𝐸
1
1
9.82 = 96.04.
CALCULATING MINIMUM SAMPLE SIZE
If you want to be 95% certain that the true population mean lies within the interval
created with an E of 1, you need AT LEAST 97 magazine advertisements in your
sample.
We round up, since 96 advertisements are not quite enough.
ASSIGNMENTS
Classwork: Pages 317-318; #2-34 Evens
Homework: Pages 317-323; #35-40 All, #45-67 Odd