Download There are 2 types of confidence intervals for the unknown mean, µ

CONFIDENCE INTERVAL WORKSHEET
MTH-1210
There are 2 types of confidence intervals for the unknown mean, µ, of a
single population. The first requires that the population standard deviation, σ, is already known (or at least bounded by a known constant);
the latter confidence interval does not require that we know σ, rather, the
the sample standard deviation, s, is used in place of σ.
For each of these confidence intervals, it is crucially important that the
sample collected is a simple random sample. The foundational theorem that
these estimates are built upon is the Central Limit Theorem. In order to use
the Central Limit Theorem reliably we must collect a simple random sample
and we must assume that the size of the sample is greater than 30 or that
the underlying population we are working with is normally distributed.
1. Confidence interval for µ when σ is known (or bounded).
For this test, we assume that σ is known (or bounded above). If σ is not
known (as is usually the case), then we must use the next confidence interval,
below. Precisely, we assume that
• We have a simple random sample
• The sample size is at least 30 or the underlying population has a normal
distribution
• The population standard deviation, σ, is already known.
A 100(1 − α)% confidence interval for the population mean µ of the population is given by,
σ
σ
x̄ − zα/2 √ , x̄ + zα/2 √
n
n
Here n is the sample size, x is the sample mean, and zα/2 is the z-value
under the normal curve for which the area under the curve, to the right of z,
is α/2. Such values can be inferred from Table
in the textbook.
1
2
CONFIDENCE INTERVAL WORKSHEET
MTH-1210
2. Confidence interval for µ when σ is unknown.
For this test, we do not need to know σ. The cost of not knowing σ is
that we now need to use the t-distribution, which depends on the number
of degrees of freedom; in this case, df = n − 1, where n is our sample size.
Precisely, we assume that
• We have a simple random sample
• The sample size is at least 30 or the underlying population has a normal
distribution
A 100(1 − α)% confidence interval for the population mean µ of the population is given by,
s
s
(x̄ − tα/2,n−1 √ , x̄ + tα/2,n−1 √ )
n
n
Here n is the sample size, x is the sample mean, s is the sample standard
deviation and tα/2,n−1 is the t-value under the t-distribution (with n−1 degrees
of freedom) curve for which the area under the curve, to the right of t, is α/2.
Such values can be inferred from Table A.5 in the book.
3. Exercises
For these exercises, σ is known. Find a 100(1−α)% confidence interval for µ
in each case. Assume that the underlying population is normally distributed.
(1) x̄ = 20, σ = 3, n = 36, α = .05
(2) x̄ = 25, σ = 3, n = 36, α = .05
(3) x̄ = 30, σ = 3, n = 25, α = .1
4. Exercises
For these exercises, σ is unknown. Find a 100(1 − α)% confidence interval for µ in each case. Assume that the underlying population is normally
distributed.
(1) x̄ = 35, s = 3.76, n = 25, α = .1
(2) x̄ = 50, s = 5.02, n = 16, α = .01
(3) x̄ = 50, s = 3.71, n = 12, α = .05
CONFIDENCE INTERVAL WORKSHEET
MTH-1210
3
5. Applied Problems
These applications use one of the types of the confidence intervals from
above. You need to determine which method is appropriate to solve these.
(1) Fifteen regions in an Amazon rain forest are randomly sampled. The
total yearly rainfalls, in centimeters, for these regions were reported as
follows:
{181, 232, 230, 158, 242, 197, 228, 302, 242, 194, 200, 186, 146, 194, 202}
Assume that yearly rainfalls within the Amazon rain forest are distributed normally. Find a confidence interval for the true mean rainfall
in the Amazon rain forest with a 95% confidence level.
(2) A simple random sample of 21 chihuahua dog weights yields a sample mean of 5.6 pounds. It is known that the standard deviation of
the population of all chihuahua weights is 1.8 pounds. Find a confidence interval for the true mean weight of all chihuahuas with a 95%
confidence level.
4
CONFIDENCE INTERVAL WORKSHEET
MTH-1210
Section 4 confidence intervals:
(1) 20 ± z.025 √336 = [19.02, 20.98]
(2) 25 ± z.025 √336 = [24.02, 25.98]
(3) 30 ± z.05 √325 = [29.01, 30.98]
Section 5 confidence intervals:
3.76
(1) 35 ± t.05,24 √
= [33.71, 36.29]
25
5.02
(2) 50 ± t.005,15 √
= [46.3, 53.7]
16
3.71
= [47.64, 52.36]
(3) 50 ± t.025,11 √
12
Solutions to the Section 6 applied problems:
(1) Since we are not given the population standard deviation, σ, we need
to compute the sample standard deviation and use the latter of our
two confidence interval recipes. We find that x̄ = 208.9 and s = 38.51.
Since we can assume that yearly rainfalls within the Amazon rain forest
are distributed normally, we can apply the t confidence interval method
despite the small sample size. Also, df = n − 1 = 14. Therefore, a
95% confidence interval for the true mean rainfall in the Congolese rain
forest is given by,
38.51
38.51
208.9 ± t.025,14 √
= 208.9 ± 2.144
= [187.582, 230.218]
3.873
15
(2) Since we are given the population standard deviation, σ = 1.8, we
us the first of our two confidence interval recipes. Therefore, a 95%
confidence interval for the true mean weight of all chihuahuas is given
by,
1.8
1.8
5.6 ± z.025 √ = 5.6 ± 1.96
= [4.83, 6.38]
4.58
21