Testing Hypothesis About Proportions, Schemes and Mind Maps of Reasoning

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Testing Hypothesis

About Proportions

Chapter 19

Objectives

• Hypothesis
• Null hypothesis
• Alternative hypothesis
• Two-sided alternative
• One-sided alternative
• P-value
• One-proportion z-test

Null and Alternative Hypothesis

1. Null Hypothesis (H

0

) – the hypothesis being

tested.

• Usually a “ no change ” or “ no difference ” statement about a parameter (mean or proportion) of the distribution.
• Example: p = p 0 (an equal sign should appear in the null hypothesis).
• Generally, it is the null hypothesis that the researcher is hoping to reject in favor of a proposed alternative hypothesis.

2. Alternative Hypothesis (H

a

or H

1

) – the

alternative to the null hypothesis.

• Often it is this hypothesis that the researcher hopes to prove true.
• Three choices possible for the alternative hypothesis.

1. If the primary concern is deciding whether a population proportion, p, is different from a specified value p 0 , the alternative hypothesis should be p≠p 0.

• Express as: Ha:p≠p 0
• A hypothesis test of this form is called a two-tailed or two- sided test.

Illustration

Summary:

Alternative Alternatives

• There are three possible

alternative hypotheses:

• H A

: parameter < hypothesized

value

• H A

: parameter ≠ hypothesized

value

• H A

: parameter > hypothesized

value

1 - Solution

• The null hypothesis is: The passing rate for teenagers is 80%, as the DMV claimed. - H 0 : p =.
• The alternative hypothesis is: The passing rate for teenagers is less than the 80% claimed by the DMV. - H a : p <.
• This hypothesis test is (single-tail) left- tailed.

2 - Choosing the Null and

Alternative Hypotheses

• Advances in medical care such as prenatal ultrasound examination now make it possible to determine a child’s sex early in pregnancy. There is a fear that in some cultures some parents may use this technology to select the sex of their children. A study for India reports that, in 1993, in one hospital, 56.9% of the live births that year were boys. It’s a medical fact that male babies are slightly more common than female babies. The study’s authors report a baseline for this region of 51.7% male live births. Is there evidence that the proportion of male births has changed? a) Determine the null hypothesis for the hypothesis test. b) Determine the alternative hypothesis for the hypothesis test. c) Classify the hypothesis test as two-tailed, left-tailed, or right-tailed.

3 - Choosing the Null and

Alternative Hypotheses

• Anyone who plays or watches sports has heard of the “home field advantage.” Teams tend to win more often when they play at home. Or do they? If there were no home field advantage, the home teams would win about half of all games played. In the 2007 MLB season, there were 2431 regular season games. It turns out that the home team won 1319 of the 2431 games, or 54.26% of the time. Could this deviation from 50% be explained just from natural sampling variability, or is it evidence to suggest that there really is a home field advantage, at least in ML baseball? a) Determine the null hypothesis for the hypothesis test. b) Determine the alternative hypothesis for the hypothesis test. c) Classify the hypothesis test as two-tailed, left-tailed, or right-tailed.

3 - Solution

• The null hypothesis is: There is no home field advantage and the proportion of home wins is 50%. - H 0 : p =.
• The alternative hypothesis is: There is a home field advantage and the proportion of home wins is greater than 50%. - H a : p >.
• This hypothesis test is (single-tail) right- tailed.

Illustration:

P-Values

• The probability that we obtain the value of the test statistic that we observed or a value that is more extreme in the direction of H a , given that H 0 is true.
• A P-value is a conditional probability.
• It is the probability of the observed test statistic given that the null hypothesis is true.
• P-value = (observed statistic value[or more extreme]|H 0 )
• The P-value is not the probability that the null hypothesis is true.

Illustration:

z 0 is the observed value of the test statistic z

Example:

Calculating P-value

• Test Statistic z = 1.71 (two-tailed)
• P-value = 2•P(z>1.71)
• P-value = 2•normalcdf(1.71,100)
• P-value =.