1. Chapter 9
Hypothesis Testing

2. Hypothesis Testing
9.1 Null and Alternative Hypotheses and Errors in Testing
9.2 z Tests about a Population Mean s Known
9.3 t Tests about a Population Mean s Unknown
9.4 z Tests about a Population Proportion
9.5 Type II Error Probabilities and Sample Size Determination (Optional)
9.6 The Chi-Square Distribution (Optional)
9.7 Statistical Inference for a Population Variance (Optional)

3. 9.1 Null and Alternative Hypotheses and Errors in Hypothesis Testing
LO 1: Specify appropriate null and alternative hypotheses.
9.1 Null and Alternative Hypotheses and Errors in Hypothesis Testing
Null hypothesis, H0, is a statement of the basic proposition being tested
Represents the status quo and is not rejected unless there is convincing sample evidence that it is false
Alternative hypothesis, Ha, is an alternative accepted only if there is convincing sample evidence it is true
One-Sided, “Greater Than” H0:   m0 vs. Ha:  > m0
One-Sided, “Less Than” H0 :   m0 vs. Ha :  < m0
Two-Sided, “Not Equal To” H0 :  = m0 vs. Ha :   m0
where m0 is a given constant value (with the appropriate units) that is a comparative value

4. Error Probabilities Type I Error: Rejecting H0 when it is true
LO 2: Describe Type I and Type II errors and their probabilities.
Error Probabilities
Type I Error: Rejecting H0 when it is true
 is the probability of making a Type I error
1 –  is the probability of not making a Type I error
Type II Error: Failing to reject H0 when it is false
β is the probability of making a Type II error
1 – β is the probability of not making a Type II error

5. 9.2 z Tests about a Population Mean: σ Known
LO 3: Use critical values and p-values to perform a z test about a
population mean when s is known.
9.2 z Tests about a Population Mean: σ Known
Test hypotheses about a population mean using the normal distribution
Called z tests
Require that the true value of the population standard deviation σ is known
In most real-world situations, σ is not known
But often is estimated from s of a single sample
When σ is unknown, test hypotheses about a population mean using the t distribution
Here, assume that we know σ

6. Steps in Testing a “Greater Than” Alternative
LO3
Steps in Testing a “Greater Than” Alternative
State the null and alternative hypotheses
Specify the significance level a
Select the test statistic
Determine the critical value rule for deciding whether or not to reject H0
Collect the sample data and calculate the value of the test statistic
Decide whether to reject H0 by using the test statistic and the rejection rule
Interpret the statistical results in managerial terms and assess their practical importance

7. 9.3 t Tests about a Population Mean: σ Unknown
LO 4: Use critical values and p-values to perform a t test about a population mean.
9.3 t Tests about a Population Mean: σ Unknown
Assume the population being sampled is normally distributed
The population standard deviation σ is unknown, as is the usual situation
If the population standard deviation σ is unknown, then it will have to estimated from a sample standard deviation s
Under these two conditions, have to use the t distribution to test hypotheses

8. Defining the t Statistic: σ Unknown
LO4
Defining the t Statistic: σ Unknown
Let x be the mean of a sample of size n with standard deviation s
Also, µ0 is the claimed value of the population mean
Define a new test statistic
If the population being sampled is normal, and s is used to estimate σ, then …
The sampling distribution of the t statistic is a t distribution with n – 1 degrees of freedom

9. t Tests about a Population Mean: σ Unknown
LO4
t Tests about a Population Mean: σ Unknown
Alternative
Reject H0 if:
p-value
Ha: µ > µ0
t > ta
Area under t distribution to right of t
Ha: µ < µ0
t < –ta
Area under t distribution to left of –t
Ha: µ  µ0
|t| > t a/2 *
Twice area under t distribution to right of |t|
tα, tα/2, and p-values are based on n – 1 degrees of freedom (for a sample of size n)
* either t > tα/2 or t < –tα/2

10. 9.4 z Tests about a Population Proportion
LO 5: Use critical values and p-values to perform a large sample z test about a population proportion.
9.4 z Tests about a Population Proportion
Alternative
Reject H0 if:
p-value
Ha: ρ > ρ0
z > za
Area under t distribution to right of z
Ha: ρ < ρ0
z < –za
Area under t distribution to left of –z
Ha: ρ  ρ0
|z| > z a/2 *
Twice area under t distribution to right of |z|
Where the test statistics is
* either z > zα/2 or z < –zα/2

11. LO 6: Calculate Type II error probabilities and
the power of a test, and determine sample size (optional).
9.5 Type II Error Probabilities and Sample Size Determination (Optional)
Want the probability β of not rejecting a false null hypothesis
That is, want the probability β of committing a Type II error
1 - β is called the power of the test
Assume that the sampled population is normally distributed, or that a large sample is taken
Test…
H0: µ = µ0 vs
Ha: µ < µ0 or Ha: µ > µ0 or Ha: µ ≠ µ0
Want to make the probability of a Type I error equal to α and randomly select a sample of size n

12. Calculating β Continued
LO6
Calculating β Continued
The probability β of a Type II error corresponding to the alternative value µa for µ is equal to the area under the standard normal curve to the left of
Here z* equals zα if the alternative hypothesis is one-sided (µ < µ0 or µ > µ0)
Also z* ≠ zα/2 if the alternative hypothesis is two-sided (µ ≠ µ0)

13. Sample Size Population is normal, or a large sample is taken
H0: m = m0 vs. Ha: m < m0 or Ha: m > m0 or Ha: m ≠ m0
Want the probability of a Type I error equal to a and probability of a Type II error corresponding to alternative value ma for m equal to b
Then take a sample of size:
z* equals za, if alternative hypothesis is one-sided and z* equals za/2 if the alternative hypothesis is two-sided
zb is point on the standard normal curve that gives a right-hand tail area equal to b

14. 9.6 The Chi-Square Distribution (Optional)
LO 7: Describe the properties of the chi-square distribution and use a chi-square table (optional).
9.6 The Chi-Square Distribution (Optional)
The chi-square ² distribution depends on the number of degrees of freedom
See Table A.17 of Appendix A
A chi-square point ²α is the point under a chi-square distribution that gives right-hand tail area 

15. 9.7 Statistical Inference for Population Variance (Optional)
LO 8: Use the chi-square distribution to make statistical inferences about population variances (optional).
9.7 Statistical Inference for Population Variance (Optional)
If s2 is the variance of a random sample of n measurements from a normal population with variance σ2
The sampling distribution of the statistic (n - 1) s2 / σ2 is a chi-square distribution with (n – 1) degrees of freedom
Can calculate confidence interval and perform hypothesis testing
100(1-α)% confidence interval for σ2