Significance Tests for Proportions

Presentation 9.2

Significance Test for p

• You can produce a confidence interval for a proportion p.

• Hypothesis tests can also be performed with one proportion to obtain evidence about the truth about a population.

Hypothesis Test Formulas

NormalcdfValuePnpp

ppZ

ppppH

ppH

a

)1(

ˆ

,,:

:

00

0

000

00Null Hypothesis

Alternate Hypothesis

Test Statistic

Remember that p0 is the null hypothesis and p-hat is the sample proportion.

In the standard error formula, we now use p0 (NOT p-hat) because our calculations are based upon the null hypothesis being true!

M&Ms Example

• The Mars Company claims that 14% of all plain candies are yellow.

• A sample from a king size bag found that 9 out of 72 candies were yellow.

• Is this significant evidence that the true proportion of yellows is not 14%?

M&Ms Example

125.072

9ˆ

14.:

14.:0

p

pH

pH

a

• Conduct a 1–proportion z-test.

• Check assumptions:– The population is greater than

10(n)=10(56)=560 so we may use the standard error formula.

– Check np>10 which is 72(.14)>10 and n(1-p)>10 which is 72(.86)>10. Since we pass these, we can approximate with the normal distribution.

• Then write hypotheses:– The proportion should be .14

(according to the company)– We suspect it may be different.

M&Ms Example

• Conduct calculations.

• Test Statistic:– Be sure to use po in

the standard error formula.

• Then calculate p:– Shown here using the

standardized data. 7138.0

)3667.,99(2

3667.0409.

015.72

)14.1(14.

14.125.

)1(

ˆ

00

0

p

NormalcdfValuep

z

z

npp

ppz

M&Ms Example

• Conclusions • With such a large p-

value, we fail to reject the null.

• There is not sufficient evidence to suggest that the proportion of yellow m&ms is not 14%.

Significance Tests for Proportions

This concludes this presentation.