1. Hypothesis Tests for a Population Mean in Practice
Chapter 10 Section 3
Hypothesis Tests for a
Population Mean in Practice

2. Chapter 10 – Section 3 Learning objectives
Test hypotheses about a population mean with σ unknown 1

3. 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
In the previous section, we assumed that the population standard deviation, σ, was known
This is not a realistic assumption
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

4. 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
The parallel between Confidence Intervals and Hypothesis Tests carries over here too
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

5. Chapter 10 – Section 3 Thus instead of the test statistic knowing σ
we calculate a test statistic using s
This is the appropriate test statistic to use when σ is unknown

6. 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
H1: μ < μ0
H1: μ > μ0

7. 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
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
The process for a hypothesis test of a mean, when σ is unknown is
Set up the problem with a null and alternative hypotheses
The process for a hypothesis test of a mean, when σ is unknown is

8. Chapter 10 – Section 3
Either the Classical and the P-value approach can be applied to determine the significance
Classical approach
P-value approach

9. 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

10. 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
An example (the same one as in Section 10.2)
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

11. 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
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

12. Chapter 10 – Section 3
Comparing using the classical approach

13. Chapter 10 – Section 3
Comparing using the P-value approach

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