Download Estimating when is unknown

Estimating 𝜇 when
𝜎 is unknown
Quantitative Methods II
Plan for Today
• Student’s t distribution
• Confidence Intervals
• Examples
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Estimating a population mean
• One of the purposes of randomly sampling a
population is to get an estimate of the mean of the
population.
• If the population standard deviation 𝜎 is not known,
then we will use a sample standard deviation s
instead.
• However, instead of the normal distribution we will
use Student’s t distribution (discovered by English
statistician William Gosset, who worked at a Guinness
brewery and published under a pseudonym).
Interval Estimates
• Interval estimate: an interval bounded by two
values that is calculated from the sample and that is
used to estimate the value of a population parameter.
• Level of confidence 1 − 𝛼 : the proportion of all
interval estimates that include the parameter being
estimated.
• Confidence interval: an interval estimate with a
specified level of confidence.
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Confidence Intervals
Summary :
Calculating Confidence Intervals
•
•
•
•
Sample Mean: 𝑥ҧ
Sample Size: n
Sample standard deviation: s
Level of confidence we wish to have: 1 − 𝛼
1 − 𝛼 ∙ 100% gives us an estimate of how
confident you can be that your mean falls within this
interval
0.95 *100% = 95%: you are 95% confident that the
population mean falls within this interval
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Student’s t distribution
Also represented by a bell-shaped curve, but it is
wider and lower than the normal curve.
The actual shape depends on the sample size,
or, degrees of freedom df = 𝑛 − 1 .
The bigger is the number of degrees of freedom,
the closer the t curve gets to the normal curve.
We have a separate table for the critical values
of Student’s t distributions.
Student’s t distribution
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Examples of table values
𝑡 7, 0.1 = 1.415
𝑡 18, 0.025 = 2.101
𝑡 25, 0.05 = 1.708
𝑡 5, 0.01 = 3.365
Step by step
Estimation of Mean μ (σ unknown)
Same assumptions as before: either the
general population has the bell-shaped
symmetric distribution, or the sample size
is at least 25.
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Confidence Coefficient 𝒕(𝐝𝐟, 𝜶Τ𝟐)
c
c
Constructing a Confidence Interval
• Step 1: Set-Up
– Describe the population parameter of interest
• Step 2: The Confidence Interval Criteria
– Check the assumptions
– Identify the probability distribution and the
formula to be used
– State the level of confidence 𝟏 − 𝜶
• Step 3: The Sample Evidence
– Collect the sample information
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Constructing a Confidence Interval
• Step 4: The Confidence Interval
– Determine the confidence coefficient
𝑡(df, 𝛼 Τ2)
– Find the error bound for a population mean
𝐸𝐵𝑀 = 𝑡(df, 𝛼 Τ2) ∙
𝑠
𝑛
– Find the lower and upper confidence limits
• Step 5: State the confidence interval
from 𝑥ҧ − 𝐸𝐵𝑀
to
𝑥ҧ + 𝐸𝐵𝑀
(units)
How to decrease the error?
• To decrease the value of EBM (and thus, to
decrease the size of the confidence interval for
𝜇) there are two possibilities:
(A) Decrease the confidence level. A smaller
confidence level will result in a smaller
𝑡(df, 𝛼/2) аnd thus, you’ll get a smaller EBM.
(B) Increase the size of a sample. A larger value of
n means a larger value of 𝑛 and thus, you’ll
get a smaller value of EBM.
• Tradeoffs: (A) less certain, (B) more costly
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Example: garbage
A sample of 11 families of four people has
produced the following amount of garbage during
a randomly selected week (in kg):
20.4 18.1 17.9 12.8 24.6 18.8 29.2 17.0 16.5
28.0 17.7
Construct a 95% confidence interval for the mean
amount of garbage produced by a family of four,
assuming that the population is normally
distributed.
Example: garbage
Step 1: What is the population parameter of
interest?
Step 2: 𝜎 is unknown, the sample size is 11, and
the population is assumed to be normally
distributed. Thus, we will be using Student’s t
distribution with 1 − 𝛼 = 0.95 (95%).
Step 3: 𝑛 = 11, and we compute 𝑥ҧ = 20.1 kg,
𝑠 = 5.0765 kg, df = 10 and 𝛼 Τ2 = 0.025.
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Example: garbage
Step 4: 𝑡(10, 0.025) = 2.228 (table)
𝐸𝐵𝑀 = 𝑡 df, 𝛼Τ2 ∙
𝑠
𝑛
= 2.228 ∙
5.0765
11
= 3.4
𝑥ҧ − 𝐸𝐵𝑀 = 16.7 , 𝑥ҧ + 𝐸𝐵𝑀 = 23.5
Step 5: The 95% confidence interval for the
population mean 𝜇 is:
from 16.7 kg to 23.5 kg
(same precision as the data)
Example: exercise
15 randomly selected students at a University
were asked how many hours they exercise per
week. Their answers are as follows:
12 1.5 3 6 4.5 10.5 15 7.5 3 10 1 8 4 3.5 5
Construct a 90% confidence interval for the mean
amount of hours of exercise per week for students
at this University, assuming that these numbers
are normally distributed.
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Example: exercise
Step 1: What is the population parameter of
interest?
Step 2: 𝜎 is unknown, the sample size is 15, and
the population is assumed to be normally
distributed. Thus, we will be using Student’s t
distribution with 1 − 𝛼 = 0.90 (90%).
Step 3: 𝑛 = 15, and we compute 𝑥ҧ = 6.3 hours,
𝑠 = 4.096 hours, df = 14 and 𝛼 Τ2 = 0.05.
Example: exercise
Step 4: 𝑡(14, 0.05) = 1.761 (table)
𝐸𝐵𝑀 = 𝑡 df, 𝛼Τ2 ∙
𝑠
𝑛
= 1.761 ∙
4.096
15
= 1.9
𝑥ҧ − 𝐸𝐵𝑀 = 4.4 , 𝑥ҧ + 𝐸𝐵𝑀 = 8.2
Step 5: The 90% confidence interval for the
population mean 𝜇 is:
from 4.4 hours to 8.2 hours
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Example: practice
Nine hotel maids in Nashville have the following
weekly incomes (in dollars):
415 565 430 510 580 450 435 505 475
Construct a 99% confidence interval for the mean
weekly income of all hotel maids in Nashville,
assuming that the incomes are normally
distributed.
Answer: from $419 to $551
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