Download Chapter 8 Section 1: Point Estimates and t-Distribution t

Chapter 8 Section 1: Point Estimates and t-Distribution
Section 1: Point Estimates
A point estimate is a one-number estimate of a parameter.
Examples:
-
The sample mean x is a point estimate for the population mean µ .
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The sample proportion p̂ is a point estimate for the population proportion p.
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The sample standard deviation s is a point estimate for the population standard deviation σ .
Example: A sample of size 1000 was taken from the USA population. The number of males in
the sample was 460. Give a point estimate for the population males’ proportion p.
Example: A sample of size 10 was taken from all Math 1530 students at APSU. Their First exam
scores were 78, 80, 55, 49, 95, 90, 77, 88, 86, 62. Give a point estimate for the mean of first
exam scores of all APSU Math 1530 students.
t - Distribution
Recall that: Central Limit Theorem indicates that for a large sample size ( n ≥ 30 ) the distribution
of the sample mean X is approximately normal with mean µ and standard deviation
σ
n
.
Problem: Usually the population mean µ and the population standard deviation σ are unknown.
Solutions:
-
In reality, we estimate the population mean µ by the sample mean x
-
We estimate the population standard deviation σ by the sample standard deviation s
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Important: In the early 1900s, a PhD student showed that when we estimate the sample standard
deviation σ with the sample standard deviation s, then the distribution of the sample mean X
follows the t – Distribution provided that
a) the sample size is large
or
b) the population is known to be normal distributed.
What is t-distribution?
Looks similar to the normal distribution but the tails are longer.
Properties of t- distribution:
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The degree of freedom (df) = n – 1 (sample size – 1)
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Total area under the curve is 1
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The curve is symmetric about 0
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As the degree of freedom (df) increases, the curve looks more like that of a standard normal
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Calculating percentile from the t-distribution:
Example: Use your calculator to calculate the following percentiles
a) 95th percentile with degree of freedom 7
b) 98th percentile with sample size n = 20
c) t (0.95,5)
d) t (0.975,10)
e) t (0.95,50)
f)
Z (0.95)
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