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Section 6-2 – Confidence Intervals for the Mean (Small Samples)
Estimating Population Parameters
The t-distribution:
What do we do if we don’t know the population standard deviation,
and can’t find a sample size of 30 or more?
If the random variable is normally distributed (or approximately
normally distributed), you can use a t-distribution.
t-distribution formula:
𝒕=
𝒙−𝝁
𝒔
𝒏
critical values of t are denoted as 𝑡𝑐 , just as critical z values are 𝑧𝑐 .
Properties of the t-distribution (Page 325)
1)
2)
3)
4)
5)
The t-distribution is bell-shaped and symmetrical about the mean.
Degrees of freedom are equal to n-1.
The total area under the curve is 1, or 100%.
The mean, median, and mode of the t-distribution are equal to
zero.
As the degrees of freedom increase, the t-distribution approaches
the normal distribution.
After 30 d.f., the t-distribution is very close to the standard normal
z-distribution.
Close enough, in fact, that we use the standard normal
distribution for d.f. ≥ 30.
We just did that in Section 6-1.
Guidelines for constructing a Confidence Interval for the Mean: tdistribution (Page 327)
1)
2)
3)
Identify the sample statistics n, 𝑥, and s.
Identify the degrees of freedom, the level of confidence c, and the
critical value 𝑡𝑐 .
Find the margin of error E.
s
𝑬 = 𝒕𝒄
; find 𝑡𝑐 by using the invT menu under 2nd Vars.
𝒏
2nd
4)
1
( (1
2
Vars 4, enter the area
− 𝑐)), and the degrees of
freedom.
The calculator will give you the critical t-value.
Find the left and right endpoints and form the confidence interval.
𝒙−𝑬<𝝁<𝒙+𝑬
The TI-84 can also do a t-distribution interval for you!!
STAT TESTS 8 (TInterval)
Same as with the z-interval, if you have the data points, select data. If
you have the statistics, select stats.
Enter the appropriate numbers and select Calculate.
EXAMPLE 1 (Page 326)
Find the critical value 𝑡𝑐 for a 95% confidence level when the sample
size is 15.
If n = 15, then the degrees of freedom are (n – 1) = 14
1
If c = .95, the area is (1 − .95) = .025 (this is the same formula we
2
used to find the area for 𝑧𝑐 in Section 6-1).
2nd VARS 4, 0.025, 14, Enter gives us a 𝑡𝑐 of 2.145 (use the positive
of the number you get).
EXAMPLE 2 (Page 327)
You randomly select 16 coffee shops and measure the temperature of
coffee sold at each. The sample mean temperature is 162.00 F with a
sample standard deviation of 10.00 F. Find the 95% confidence interval
for the mean temperature. Assume the temperatures are approximately
normally distributed.
We MUST use the t-distribution for this; the sample size is less than
30 and we don’t know what σ is, but we do know that the distribution
is approximately normal.
STAT TESTS 8, select Stats, and enter the values for 𝑥, s, n, and c.
The 95% confidence interval is from 156.7 to 167.3.
We round to one decimal because we were given 𝑥 rounded to
one decimal.
SO, we are 95% confident that the actual population mean of coffee
temperature in ALL coffee shops is between 156.70F and 167.30F.
EXAMPLE 3 (Page 328)
You randomly select 20 mortgage institutions and determine the
current mortgage interest rate at each. The sample mean rate is 6.22%,
with a sample standard deviation of 0.42%. Find the 99% confidence
interval for the population mean mortgage interest rate. Assume the
interest rates are approximately normally distributed.
We MUST use the t-distribution for this; the sample size is less than
30 and we don’t know what σ is, but we do know that the distribution
is approximately normal.
STAT TESTS 8, select Stats, and enter the values for 𝑥, s, n, and c.
The 99% confidence interval is from 5.95 to 6.49.
We round to two decimals because we were given 𝑥 rounded to
two decimals.
SO, we are 99% confident that the actual population mean mortgage
interest rate for ALL mortgage institutions is between 5.95% and
6.49%.
Now, find the 90% and 95% confidence intervals for the population
mean mortgage interest rate.
What happens to the widths of the intervals as the confidence
levels change?
The more confident we need to be, the wider the interval must
become.
Put another way, the higher c is, the wider the interval will
be.
There is a flow chart on how to decide which distribution to use (t or
normal) on page 329. Study this!!
Is n ≥ 30?
Use the normal distribution
s
with 𝐸 = 𝒛𝒄 ; if σ is
YES
𝒏
unknown, use s instead.
NO
Is the population normally,
or approximately normally,
distributed?
NO
You can NOT use the normal
or the t-distribution.
YES
Is σ known?
Use the normal distribution
s
with 𝐸 = 𝒛𝒄 𝒏
YES
NO
Use the t-distribution with 𝐸 = 𝒕𝒄
and n – 1 degrees of freedom.
s
𝒏
EXAMPLE 4 (Page 329)
You randomly select 25 newly constructed houses. The sample mean
construction cost is $181,000 and the population standard deviation is
$28,000. Assuming construction costs are normally distributed, should
you use the normal distribution, the t-distribution, or neither to
construct a 95% confidence interval for the population mean
construction costs? Explain your reasoning.
Although n is less than 30, we can still use the normal distribution
because we know that the distribution is normal and we know what σ
is.
You randomly select 18 adult male athletes and measure the resting heart
rate of each. The sample mean heart rate is 64 beats per minute with a
sample standard deviation of 2.5 beats per minute. Assuming the heart
rates are normally distributed, should you use the normal distribution,
the t-distribution, or neither to construct a 90% confidence interval for
the mean heart rate? Explain your reasoning.
Because n < 30, the distribution is normal, and we do not know what
σ is, we should use the t-distribution on this one.
ASSIGNMENTS
Classwork: Page 330; #1-16 All
Homework: Pages 331-332; #17-28