Download estimate ± margin of error

Confidence Intervals Day 2
Last class we saw "what" a confidence interval is.
We know that an 80% confidence level means that 80 out of 100 samples
will produce an interval that contains the actual parameter we are interested
in finding.
We also saw that all confidence intervals have the basic form:
estimate±marginoferror
We can rewrite this as the following:
estimate±(criticalvalue)(standarddeviationofstatistic)
or
Example 1: zc can be found using our z-score table.
a) Find zc when c = 80
b) Find zc when c = 95
A quick reference for common zc
Level of Confidence
zc
80
____________
90
1.645
95
_____________
99
2.575
Remember using table A to find critical
values (z-scores) only works if x-bar is
NORMAL! Don't forget about the CLT.
Example 2: Myface, a social networking site, allows its users to add friends, send
messages, and update their personal profiles. The following represents a random
sample of the number of friends for 40 users of the website. Assume that the standard
deviation is σ = 57 users.
140
105
130
97
80
165
232
110
214
201
122
98
65
88
154
133
121
82
130
211
153
114
58
77
51
247
236
109
126
132
125
149
122
74
59
218
192
90
117
105
a) Calculate a point estimate for the number of friends a Myface user has on average.
b) Use a 95% confidence level to find the margin error for the mean
number of friends of all users of the myface website.
c) Construct a confidence interval using part (b) and (c)
d) Interpret what the confidence interval means.
e) Create confidence interval using a 99% confidence level
f) Compare the widths of part (e) and (c)
Example 3: A random sample of 34 home theater systems has a
mean price of $452.80. The standard deviation for all home theater
systems is $85.50. Construct a 94% confidence interval and interpret the
results.
Example 4: A random sample of 45 soccer coaches found the mean number
of years coaching to be 6.8 years. The standard deviation for all soccer
coaches is 4.7 years. Construct a 90% confidence interval and interpret the
results.
Calculating a Required Sample Size
By using the margin of error formula,
we can calculate the required sample size to reach a
certain margin of error.
You will often have an idea of the highest error that you will want and will
be okay with a smaller margin of error so we'll write it as an inequality
Example 5:
Researchers would like to estimate the mean cholesterol level, μ, of a particular variety of
monkey that is often used in laboratory experiments. They would like their estimate to be
within 1 milligram per deciliter (mg/dl) of the true value of μ at a 95% confidence level.
A previous study involving this variety of monkey suggests that the standard deviation of
cholesterol level is about 5 mg/dl.
Obtaining monkeys is time consuming and expensive, so the researchers want to know
the minimum number of monkeys they will need to generate a satisfactory estimate.
Example 6:
A restaurant owner wishes to find the 92% confidence interval of
the true mean cost of a meal for a costumer. How large should the
sample be if she wishes to be accurate within $0.10? A previous study
showed that the standard deviation of the price was $1.25.
Example 7:
You want to estimate the mean number of friends for all users on Myface.
How many users must be included in the sample if you want to be 98%
confident that the sample mean is within 7 friends of the population mean?
(remember we were given the standard deviation in example 2)