Section 3.3: The Story of Statistical InferenceSection 4.1: Testing Where a Proportion Is

Statistical Inference: Confidence Intervals vs. Hypothesis Testing

A confidence interval is used to estimate an unknown population parameter, e.g. p.

A hypothesis test is used to test a claim about a population parameter, e.g. p.

Determining the Child’s Sex During Pregnancy

Advances in medicine make it possible to determine the sex of a child early in a pregnancy. Because some cultures value male children more highly than female children, there’s a fear that some parents may not carry pregnancies of girls to term.

Punjab, India Study – 1994

56.9% of the 550 live births that year were boys. It’s a medical fact that male babies are slightly

more common than female babies. The authors report a baseline for this region of 51.7% male live births.

Question: Is the sample proportion of 56.9% evidence of a higher proportion of male births?

The Nuts and Bolts

Who is the population?

What is the parameter of interest?

Facts about the Null Hypothesis, Recall that is a statement about the

population parameter, p, and not the sample statistic, .

is the hypothesis of “no difference”.

While performing the hypothesis test we assume the null hypothesis is true and see if we can find enough evidence to disprove this claim.

0H

0H

p̂

0H

Testing this Claim

Collect a random sample from the population and compute the sample proportion.

Statistical Significance

To determine if our sample results are statistically significant, we need to determine the p-value:

Assuming the null hypothesis is correct,

how likely is it to get a sample proportion as extreme or more extreme than the one that we observed?

To answer this question, we need to know how varies in repeated sampling.

Easy question, right? Draw a well-labeled graph which describes

how repeated random samples, each of size 550, would vary.

p̂

Intuition Check

Before doing any calculations and by looking at the graph you drew, do you think there’s evidence to suggest that birth ratios of boys to girls is equal?

Let’s now test your intuition.

How close is to p?

The z-value provides us the answer since it is a measure of how many standard errors our sample statistic is from our population parameter.

Compute the z-value and interpret it in the context of this problem.

..

ˆ

es

ppz

p̂

More on the z-value

A sample proportion’s z-value indicates where, in the distribution of sample values, that proportion falls.

What does a negative z-value tell you? What does a positive z-value tell you? What does a z-value of -0.85 tell you? What does a z-value of 5.7 tell you?

Finding the P-value

See Table 4.1.1. in the text.

Conclusions

Is there enough evidence to reject the null hypothesis or should we fail to reject the null hypothesis? (Note: We do not ever accept the null hypothesis.)

Legal system analogy.

Example 2:

Suppose instead that 52.6% of the 550 live births were male. Would this sample proportion have been strong enough to reject the null hypothesis?

Do the appropriate calculations.

How small does the p-value need to be to reject the null hypothesis? What if ?

What if ?

What if ?

How about if ?

000,000,10

570000057.0 p

000,10

240024.0 p

4

1

100

2828.0 p

100

404.0 p

Level of Significance,

We can define “rare event” arbitrarily by setting a threshold ( -value) for our p-value.

If the p-value falls below the threshold, we’ll reject the null hypothesis and call the results statistically significant. If not we fail to reject the null hypothesis.