Download Exercise 3 - OCVTS MATES-STAT

NAME:_________________________________BLOCK:___________DATE:________
8.1 Estimating µ when σ is known
A point estimate is a single value estimate for a population parameter. The most unbiased
point estimate of the population mean μ is the sample mean 𝑥. The margin of error is the
greatest possible distance between the point estimate and the value of the parameter.
Exercise 1: A social networking website 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. Find a point estimate of the population
mean, µ. (Source: Facebook)
140
122
153
125
105
98
114
149
130
65
58
122
Point Estimate: 𝑥 =
97
88
51
74
∑𝑥
𝑛
80
154
77
59
165
133
247
218
232
121
236
192
110
82
109
90
214
130
126
117
=
Exercise 2: Use the social networking website data and a 95% confidence level to find
the margin of error for the mean number of friends for all users of the website. Assume
that the population standard deviation is about 53.0.
Critical Values [Page A23]
𝜎
𝐸 = 𝑧𝑐 𝜎𝑥 = 𝑧𝑐 𝑛 =
𝑧𝑐 = 1.96
Interpretive Statement
We are ___ confident that the margin of error for the
population mean is about ___ friends.
√
Confidence Interval: A c-confidence interval for the population mean μ is given by:
𝜎
𝑥 − 𝐸 < 𝜇 < 𝑥 + 𝐸 where 𝐸 = 𝑧𝑐 𝑛
√
The probability that the confidence interval contains μ is c.
Exercise 3: Construct a 95% confidence interval for the mean number of friends for all
users of the website.
Left Endpoint: 𝑥 − 𝐸
Right Endpoint: 𝑥 + 𝐸
Confidence Interval
1
201
211
132
105
NAME:_________________________________BLOCK:___________DATE:________
Exercise 4: A college admissions director wishes to estimate the mean age of all
students currently enrolled. In a random sample of 20 students, the mean age is found to
be 22.9 years. From past studies, the standard deviation is known to be 1.5 years, and the
population is normally distributed. Construct a 90% confidence interval of the population
mean age.
Critical Values
𝐸 = 𝑧𝑐 𝜎𝑥 = 𝑧𝑐
𝜎
√𝑛
𝑧𝑐 = 1.645
=
Left Endpoint: 𝑥 − 𝐸
Right Endpoint: 𝑥 + 𝐸
Confidence Interval
Interpretive Statement
With 90% confidence, we can say that the mean age of
all the students is between ____ and ____years.
TI-84: ZInterval Function
The ZInterval (STAT / TESTS / 7: ZInterval) function computes a confidence interval for
and unknown population mean µ when the population standard deviation is known.
Inputs
Output
Margin of Error
σ, 𝑥, n, 𝑐 − 𝑙𝑒𝑣𝑒𝑙
𝑥 − 𝐸 𝑡𝑜 𝑥 + 𝐸
1
𝐸 = (𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ)
2
Exercise 5: Use the TI-84 calculator to solve Exercise 4.
Inputs
Output
Margin of Error
Minimum Sample Size: Given a c-confidence level and a margin of error E, the
minimum sample size n needed to estimate the population mean µ is given by
𝑧𝑐 𝜎 2
𝑛= (
)
𝐸
If n is not a whole number, increase n to the next higher whole number.
2
NAME:_________________________________BLOCK:___________DATE:________
Exercise 6: You want to estimate the mean number of friends for all users of the
website. How many users must be included in the sample if you want to be 95%
confident that the sample mean is within seven friends of the population mean? Assume
the population standard deviation is about 53.0.
Critical Values [Page A23]
𝑧𝑐 𝜎 2
𝑛= (
) =
𝐸
Apply Ceiling Function
Interpretive Statement
𝑧𝑐 = 1.96
Exercise 7: A sample of 50 salmon is caught and weighed. The sample standard
deviation of the 50 weights is 2.15 lb. How large of a sample should be taken to be 97%
confident that the sample mean is within 0.20 lb of the mean weight of the population?
Summarize your results in a complete sentence relevant to this application.
Critical Values
𝑧𝑐 𝜎 2
) =
𝐸
𝑛= (
Apply Ceiling Function
Interpretive Statement
A sample size of ____ salmon will be large enough to
ensure that we are 97% confident that the sample mean
is within _____ of the population mean.
8.2 Estimating µ when σ is Unknown
3
NAME:_________________________________BLOCK:___________DATE:________
Exercise 1: Find the critical value 𝒕𝒄 for a 95% confidence level when the sample size is
15.
𝑑𝑓 = 𝑛 − 1
𝑑𝑓 = 15 − 1 = 14
𝑡𝑐 =
𝑡𝑐 = 2.145
Exercise 2: You randomly select 16 coffee shops and measure the temperature of the
coffee sold at each. The sample mean temperature is 162.0ºF with a sample standard
deviation of 10.0ºF. Find the 95% confidence interval for the population mean
temperature. Assume the temperatures are approximately normally distributed.
𝑛=
𝑑𝑓 = 𝑛 − 1
𝑥=
𝑠=
𝑡𝑐 =
𝑠
𝐸 = 𝑡𝑐 𝑛
√
Left Endpoint: 𝑥 − 𝐸
Right Endpoint: 𝑥 + 𝐸
Confidence Interval
Exercise 3: An archeologist discovered a new, but extinct, species of miniature horse.
The only seven known samples show shoulder heights (in cm) of 45.3, 47.1, 44.2, 46.8,
46.5, 45.5, and 47.6. Assume that the population of shoulder heights is approximately
normal. Find the 99% confidence interval for the mean height of the entire population of
ancient horses and the margin of error E. Then summarize your results in a complete
sentence relevant to this application.
𝑛=
𝑑𝑓 = 𝑛 − 1
𝑥=
𝑠=
𝑡𝑐 =
𝑠
𝐸 = 𝑡𝑐 𝑛
√
Left Endpoint: 𝑥 − 𝐸
Right Endpoint: 𝑥 + 𝐸
Confidence Interval
Interpretive Statement
The archaeologist can be 99% certain that the interval from
_____ cm to _____ cm contains the population mean µ for
shoulder height of this species of miniature horse.
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NAME:_________________________________BLOCK:___________DATE:________
The TInterval (STAT / TESTS / 8: TInterval) function computes a confidence interval
for the population mean µ. When the population standard deviation is unknown, it is
approximated by the sample standard deviation s. The TInterval function works with the
Student’s t-distribution.
Inputs
Output
Margin of Error
s, 𝑥, n, 𝑐 − 𝑙𝑒𝑣𝑒𝑙
𝑥 − 𝐸 𝑡𝑜 𝑥 + 𝐸
1
𝐸 = (𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ)
2
Exercise 4: Repeat Exercise 3. Use the TInterval function.
Inputs
s = 1.19, 𝑥 = 22.9, 𝑛 = 7, 𝑐 = 0.99
Output
(44.5,47.8)
1
Margin of Error
𝐸 = (47.8 − 44.5) = 1.67
2
8.3 Estimating 𝒑 in the Binomial Distribution
Confidence Interval: A c-confidence interval for the population proportion p is given
by:
𝑝̂𝑞̂
𝑝̂ − 𝐸 < 𝑝 < 𝑝̂ + 𝐸 where 𝐸 = 𝑧𝑐 √ 𝑛
Exercise 1: In a survey of 1000 U.S. adults, 662 said that it is acceptable to check
personal e-mail while at work. Construct a 95% confidence interval for the population
proportion of U.S. adults who say that it is acceptable to check personal e-mail while at
work.
Critical Values
𝑛=
𝑝̂ = r/n
𝑞̂ = 1 − 𝑝̂ =
𝑛𝑝̂ =
𝑛𝑞̂ =
𝑝̂𝑞̂
𝐸 = 𝑧𝑐 √ 𝑛 =
Left Endpoint: 𝑝̂ − 𝐸
Right Endpoint: 𝑝̂ + 𝐸
Confidence Interval
Interpretive Statement
5
NAME:_________________________________BLOCK:___________DATE:________
Exercise 2: Suppose that 800 students were selected at random from a student body of
20,000 and given flu shots. All 800 students were exposed to the flu, and 600 of them did
not get the flu. Let p represent the probability that the shot will be successful for any
single student selected at random from the entire population. Find a 99% confidence
interval for p.
Critical Values
𝑛=
𝑝̂ = r/n
𝑞̂ = 1 − 𝑝̂ =
𝑛𝑝̂ =
𝑛𝑞̂ =
𝑝̂𝑞̂
𝐸 = 𝑧𝑐 √ 𝑛 =
Left Endpoint: 𝑝̂ − 𝐸
Right Endpoint: 𝑝̂ + 𝐸
Confidence Interval
Interpretive Statement
The 1-PropZInterval function computes a confidence interval for the population
proportion p.
Exercise 3: Re-compute the solution to Exercise 2 using the 1-PropZInterval function.
Inputs
Output
Margin of Error
𝑥 = 600, 𝑛 = 800, 𝑐 = 0.99
(0.711,0.789)
1
𝐸 = (0.789 − 0.711) = 0.039
2
Minimum Sample Size n for Estimating a Proportion p
𝑧𝑐 2
𝑛 = 𝑝𝑞 ( ) (𝑤𝑖𝑡ℎ 𝑝𝑟𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑟𝑦 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑓𝑜𝑟 𝑝)
𝐸
1 𝑧𝑐 2
𝑛 = ( ) (𝑛𝑜 𝑝𝑟𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑟𝑦 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑓𝑜𝑟 𝑝)
4 𝐸
6
NAME:_________________________________BLOCK:___________DATE:________
Exercise 4: You are running a political campaign and wish to estimate, with 95%
confidence, the population proportion of registered voters who will vote for your
candidate. Your estimate must be accurate within 3% of the true population proportion.
Find the minimum sample size needed if
a) No preliminary estimate for p is available
Critical Values
𝐸=
1 𝑧
2
𝑛 = 4 ( 𝐸𝑐) =
Apply Ceiling Function
Interpretive Statement
b) A preliminary estimate for p is 0.31
Critical Values
𝐸=
𝑧𝑐 2
𝑛 = 𝑝𝑞 ( ) =
𝐸
Apply Ceiling Function
Interpretive Statement
Exercise 5: You wish to estimate, with 90% confidence and within 2% of the true
population, the proportion of adult’s age 18 to 29 who have high blood pressure. Find the
minimum sample size needed if
a) No preliminary estimate for p is available
Critical Values
𝐸=
1 𝑧
2
𝑛 = ( 𝑐) =
4
𝐸
Apply Ceiling Function
Interpretive Statement
7
NAME:_________________________________BLOCK:___________DATE:________
b) A preliminary estimate for p found that 4% of adults in this age group had high blood
pressure
Critical Values
𝐸=
𝑧𝑐 2
𝑛 = 𝑝𝑞 ( ) =
𝐸
Apply Ceiling Function
Interpretive Statement
c) Determine how many more adults are needed when no preliminary estimate is
available compared to when a preliminary estimate for p is 4%.
1692 − 260 = 1432
8