Transcript MATH 1410/6.1 and 6.2 pp

CHAPTER SIX
Confidence
Intervals
Section 6.1
Confidence Intervals for the
MEAN (Large Samples)
Estimating Vocab
 Point Estimate: a single value estimate for a
population parameter.
 Interval Estimate: a range of values used to
estimate a population parameter.
 Level of Confidence (c): the probability that the
interval estimate contains the population
parameter
 The level of confidence, c, is the area under the
curve between 2 z-scores called Critical Values
Find the critical value zc necessary
to construct a confidence interval
at the given level of confidence.
C = 0.85
C = 0.75
More Vocab!

Find the margin of error for the
given values.
 C = 0.90
s = 2.9 n = 50
 C = 0.975
s = 4.6 n = 100
Confidence Intervals for the
Population Mean

Construct a C.I. for the Mean
 1. Find the sample mean and sample size.
 2. Specify σ if known. Otherwise, if n > 30
, find the sample standard deviation s.
 3. Find the critical value zc that
corresponds with the given level of
confidence.
 4. Find the margin of error, E.
 5. Find the left and right endpoints and
form the confidence interval.
Construct the indicated confidence
interval for the population mean.
 44. A random sample of 55 standard hotel rooms
in the Philadelphia, PA area has a mean nightly
cost of $154.17 and a standard deviation of
$38.60. Construct a 99% confidence interval for
the population mean. Interpret the results.
 46. Repeat Exercise 44, using a standard
deviation of s = $42.50. Which confidence
interval is wider? Explain.
Sample Size: given c and E…
Example from p 308
 54. A soccer ball manufacturer wants to estimate
the mean circumference of mini-soccer balls within
0.15 inch. Assume the population of circumferences
is normally distributed.
 A) Determine the minimum sample size required to
construct a 90% C.I. for the population mean.
Assume σ = 0.20 inch.
 B) Repeat part (A) using σ = 0.10 inch.
 C) Which standard deviation requires a larger
sample size? Explain.
Section 6.2
Confidence Intervals for the
MEAN (Small Samples)
The t – Distribution (table #5)
 Used when the sample size n < 30 , the
population is normally distributed, and σ
is unknown.
 t – Distribution is a family of curves.
 Bell shaped, symmetric about the mean.
 Total area under the t - curve is 1
 Mean, median, mode are equal to 0
 Uses Degrees of Freedom (d.f. =n–1)
 d. f. are the # of free choices after a the
sample mean is calculated.
 To find the critical value, tc , use the t
table.
 Find the critical value, tc for c = 0.98, n = 20
 Find the critical value, tc for c = 0.95, n = 12
Confidence Intervals and
t - Distributions
 1. Find the sample mean, standard
deviation, and sample size.
 2. ID the degrees of freedom, level of
confidence and the critical value.
 3. Find the margin of error, E.
 4. Find the left and right endpoints and
for the confidence interval.
Construct the indicated C.I.
Use a Normal or a t – Distribution to
construct a 95% C.I. for the population
mean. (from page 317)
 36. In a random sample of 13 people,
the mean length of stay at a hospital
was 6.2 days. Assume the population
standard deviation is 1.7 days and the
lengths of stay are normally
distributed.
 Find the sample mean and standard
deviation, the construct a 99% C.I. for the
population mean. (from p 316)
 28. The weekly time spent (in hours) on
homework for 18 randomly selected high
school students:
12.0 11.3 13.5 11.7 12.0 13.0
15.5 10.8 12.5 12.3 14.0 9.5
8.8 10.0 12.8 15.0 11.8 13.0