1. Testing Hypotheses about a Population Proportion
Lecture 41
Sections 9.1 – 9.3
Tue, Apr 6, 2004

2. Learning about a Population Parametr
When we ask questions about a population parameter, our questions generally fall into two categories.
What is the value of the parameter?
That is, estimate its value.
Does the evidence tend to support or refute a claim about the parameter?
That is, test a hypothesis concerning the parameter.

3. Testing Hypotheses about a Population Parameter
Example: A customary assumption is that a newborn baby is as likely to be a boy as it is to be a girl.
However, some believe that boys are move likely.
A random sample of 1000 live births shows that 520 are boys and 480 are girls.
Does this evidence support or refute the customary assumption?

4. The Steps of Testing a Hypothesis
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.

5. 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.

6. 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.
H1: p  p0.
H1: p > p0.

7. Explaining the Data
The observation is 520 males out of 1000 births, or 52%.
That is not 50%.
How do we explain the discrepancy?
Chance.
The true proportion is not 50%, but something larger, maybe 52%.

8. 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.

9. The Sampling Distribution of p^
To decide whether the sample evidence is significant, we will compare the p-value to .
 is based on a critical value.
 is the probability that the sample data are at least as extreme as the critical value, if the null hypothesis is true.

10. The Sampling Distribution of p^
Therefore, when we compute the p-value, we do it under the assumption that H0 is true.
In other words, for the sake of calculating the p-value, we should assume that p = p0.

11. The Sampling Distribution of p^
We know that the sampling distribution of p^ is normal with mean p and standard deviation
p^ = (p(1 – p)/n).
Thus, we assume that p^ has mean p0 and standard deviation
p^ = (p0(1 – p0)/n).

12. The Test Statistic
The test statistic is the z-score of p^, under the assumption that H0 is true.
Thus,
Z = (p^ – p0)/p^.
In our example, we have
p0 = 0.50.
p^ = (0.50(1 – 0.50)/1000) =

13. The Test Statistic Therefore, the test statistic is
Z = (p^ – 0.50)/0.0158
Now all we need to do to get the value of the test statistic is to collect the sample data and substitute the value of p^.

14. Computing the Test Statistic
In the sample, p^ = 0.52.
Thus,
z = (0.52 – 0.50)/0.0158
= 1.26.

15. Compute the p-value
To compute the p-value, we must 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.

16. Compute the p-value In our example, the test is one-tailed.
So we compute
P(Z > 1.26) =
I found this by evaluating
normalcdf(1.26, 99)
on the TI-83.

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

18. 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.”

19. 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.

20. 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.

21. 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.
p^.
The sample size.
We are interested in the p-value.

22. Assignment Page 534: Exercises 1 – 17.
If the problem says to show all of the steps, be sure that you show all of the steps.
Show the steps clearly and in the proper order.