1. Testing Hypotheses about a Population Proportion
Lecture 31
Sections 9.1 – 9.3
Wed, Mar 16, 2005

2. Discovering Characteristics of a Population
Any question about a population must first be described in terms of a population parameter.
Then the question about that parameter generally falls into one of two categories.
What is the value of the parameter?
That is, estimate its value.
Does the evidence support or refute a claim about the value of the parameter?
That is, test a hypothesis concerning the parameter.

3. Example
A standard assumption is that a newborn baby is as likely to be a boy as to be a girl. However, some people believe that boys are more likely.
Suppose a random sample of 1000 live births shows that 520 are boys and 480 are girls.
Use the data to estimate the proportion of male births.
Does this evidence support or refute the standard assumption?

4. Two Approaches for Hypothesis Testing
Classical approach.
Determine the critical value and the rejection region.
See whether the statistic falls in the rejection region.
Report the decision.
p-Value approach.
Compute the p-value of the statistic.
See whether the p-value is less than the significance level.
Report the p-value.

5. Classical Approach 

6. Classical Approach  

7. Classical Approach   z c

8. Classical Approach   z c
Acceptance Region
Rejection Region

9. Classical Approach  
Reject z c z
Acceptance Region
Rejection Region

10. Classical Approach 
Accept  z z c
Acceptance Region
Rejection Region

11. p-Value Approach

12. The Steps of Testing a Hypothesis (p-Value Approach)
The basic steps are
1. State the null and alternative hypotheses.
2. State the significance level.
3. Compute the value of the test statistic.
4. Compute the p-value.
5. State the conclusion.
See page 519 (I omitted the first step.)

13. Step 1: State the Null and Alternative Hypotheses
Let p = proportion of live births that are boys.
The null and alternative hypotheses are
H0: p = 0.50.
H1: p > 0.50.

14. State the Null and Alternative Hypotheses
The null hypothesis should state a hypothetical value p0 for the population proportion.
H0: p = p0.
The alternative hypothesis must contradict the null hypothesis in one of three ways:
H1: p < p0. (Direction of extreme is left.)
H1: p  p0. (Direction of extreme is left and right.)
H1: p > p0. (Direction of extreme is right.)

15. Explaining the Data
The observation is 520 males out of 1000 births, or 52%. That is, p^ = 0.52.
Since we did not observe 50%, how do we explain the discrepancy?
Chance, or
The true proportion is not 50%, but something larger, maybe 52%.

16. Step 2: State the Significance Level
The significance level  should be given in the problem.
If it isn’t, then use  = 0.05.
In this example, we will use  = 0.05.

17. The Sampling Distribution of p^
To decide whether the sample evidence is significant, we will compare the p-value to .
From the value of , we may find the critical value(s).
 is the probability that the sample data are at least as extreme as the critical value(s), if the null hypothesis is true.

18. The Sampling Distribution of p^
Therefore, when we compute the p-value, we do it under the assumption that H0 is true, i.e., that p = p0.

19. The Sampling Distribution of p^
We know that the sampling distribution of p^ is normal with mean p and standard deviation
Thus, we assume that p^ has mean p0 and standard deviation:

20. Step 3: The Test Statistic
Test statistic – The z-score of p^, under the assumption that H0 is true.
Thus,

21. The Test Statistic In our example, we compute
Therefore, the test statistic is
Now, to find the value of the test statistic, all we need to do is to collect the sample data and substitute the value of p^.

22. Computing the Test Statistic
In the sample, p^ = 0.52.
Thus,
z = (0.52 – 0.50)/ = 1.26.

23. Step 4: Compute the p-value
To compute the p-value, we must first check whether it is a one-tailed or a two-tailed test.
We will compute the probability that Z would be at least as extreme as the value of our test statistic.
If the test is two-tailed, then we must take into account both tails of the distribution to get the p-value.

24. Compute the p-value
In this example, the test is one-tailed, with the direction of extreme to the right.
So we compute
P(Z > 1.26) =

25. Compute the p-value An alternative is to evaluate
normalcdf(0.52, E99, 0.50, )
on the TI-83.
It should give the same answer (except for round-off).

26. Step 5: State the Conclusion
Since the p-value is greater than , we should not reject the null hypothesis.
State the conclusion in a sentence.
“The data do not support the claim, at the 5% level of significance, that more than 50% of live births are male.”

27. Testing Hypotheses on the TI-83
The TI-83 has special functions designed for hypothesis testing.
Press STAT.
Select the TESTS menu.
Select 1-PropZTest…
Press ENTER.
A window with several items appears.

28. Testing Hypotheses on the TI-83
Enter the value of p0. Press ENTER and the down arrow.
Enter the numerator x of p^. Press ENTER and the down arrow.
Enter the sample size n. Press ENTER and the down arrow.
Select the type of alternative hypothesis. Press the down arrow.
Select Calculate. Press ENTER.
(You may select Draw to see a picture.)

29. Testing Hypotheses on the TI-83
The display shows
The title “1-PropZTest”
The alternative hypothesis.
The value of the test statistic Z.
The p-value.
The value of p^.
The sample size.
We are interested in the p-value.