Download 10.3

Chapter 10
Section 3
Hypothesis Tests for a
Population Mean in Practice
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 1 of 14
Chapter 10 – Section 3
● Learning objectives
1

Test hypotheses about a population mean with σ
unknown
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 2 of 14
Chapter 10 – Section 3
● In the previous section, we assumed that the
population standard deviation, σ, was known
● This is not a realistic assumption
● There is a parallel between Chapters 9 and 10
 Sections 9.1 and 10.2 … solving the problems
assuming that σ was known
 Sections 9.2 and 10.3 … solving the problem
assuming that σ was not known
● σ not being known is a much more practical
assumption
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 3 of 14
Chapter 10 – Section 3
● The parallel between Confidence Intervals and
Hypothesis Tests carries over here too
● For Confidence Intervals
 We estimate the population standard deviation σ by
the sample standard deviation s
 We use the Student’s t-distribution with n-1 degrees
of freedom
● For Hypothesis Tests, we do the same
 Use σ for s
 Use the Student’s t for the normal
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 4 of 14
Chapter 10 – Section 3
● Thus instead of the test statistic knowing σ
x  0
z0 
/ n
we calculate a test statistic using s
x  0
t
s/ n
● This is the appropriate test statistic to use when
σ is unknown
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 5 of 14
Chapter 10 – Section 4
● We can perform our hypotheses for tests of a
population proportion in the same way as when
the sample standard deviation is known
Two-tailed
Left-tailed
Right-tailed
H0: μ = μ0
H1: μ ≠ μ0
H0: μ = μ0
H1: μ < μ0
H0: μ = μ0
H1: μ > μ0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 6 of 14
Chapter 10 – Section 3
● The process for a hypothesis test of a mean,
when σ is unknown is
 Set up the problem with a null and alternative
hypotheses
 Collect the data and compute the sample mean
 Compute the test statistic
x  0
t
s/ n
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 7 of 14
Chapter 10 – Section 3
● Either the Classical and the P-value approach
can be applied to determine the significance
Classical approach
P-value approach
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 8 of 14
Chapter 10 – Section 3
● There are thus only differences between this
process and the one using the normal
distribution, covered in Section 10.2
 We use the sample standard deviation s instead of
the population standard deviation σ
 We use the Student’s t-distribution, with n-1 degrees
of freedom, instead of the normal distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 9 of 14
Chapter 10 – Section 3
● An example (the same one as in Section 10.2)
● A gasoline manufacturer wants to make sure
that the octane in their gasoline is at least 87.0
 The testing organization takes a sample of size 40
 The sample standard deviation is 0.5
 The sample mean octane is 86.94
● Our null and alternative hypotheses
 H0: Mean octane = 87
 HA: Mean octane < 87
 α = 0.05
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 10 of 14
Chapter 10 – Section 3
● Do we reject the null hypothesis?




86.94 is 0.06 lower than 87.0
The standard error is (0.5 / √ 40) = 0.08
0.06 is 0.75 standard error less
The critical t value, with 39 degrees of freedom, is
1.685
 –1.685 < –0.75, it is not unusual
● Our conclusion
 We do not reject the null hypothesis
 We have insufficient evidence that the true population
mean (mean octane) is less than 87.0
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 11 of 14
Chapter 10 – Section 3
● Comparing using the classical approach
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 12 of 14
Chapter 10 – Section 3
● Comparing using the P-value approach
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 13 of 14
Summary: Chapter 10 – Section 3
● A hypothesis test of means, with σ unknown, has
the same general structure as a hypothesis test
of means with σ known
● Any one of our three methods can be used, with
the following two changes to all the calculations
 Use the sample standard deviation s in place of the
population standard deviation σ
 Use the Student’s t-distribution in place of the normal
distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 10 Section 3 – Slide 14 of 14