Warm UP• In 2000, Florida changed its motorcycle helmet law.

No longer are riders 21 and over required to wear helmets, but those under 21 must wear helmets. Police reports of motorcycle accidents record whether the rider wore a helmet and the rider’s age. Before the change in law, 60% of youths involved in an accident had been wearing their helmets. After the law changed police observed that out of 781 young riders involved in accidents, 396 (50.7%) were wearing helmets. Has helmet use in Florida among riders under 21 declined due to the change in law? Perform a formal hypothesis test.

Answer on pg. 482

Ch. 22 – Comparing Two Proportions(Day 2 – Hypothesis Tests)

Part V – From the Data at Hand to the World at Large

Prospective Study• In Michigan, the High/Scope Educational Research Foundation

followed two groups of low-income children from early childhood through adulthood. Their goal was to determine the effects of pre-school on the future success of poor children.

• The researchers followed 62 children who attended preschool as 3- and 4-year olds. Of this group, 38 needed social services (mainly welfare) as adults.

• They followed 61 children who did not attend preschool. Of this group, 49 needed social services (mainly welfare) as adults.

• Does this study provide evidence that children who attend preschool have less need for social services as adults than those who don’t?

Comparing the proportions• First, let’s compare the two sample proportions:

• The samples do show a difference, but is it a big enough difference to be statistically significant?

• We have to perform a hypothesis test to answer this question

1

2

38ˆ .6136249ˆ .80361

p

p

Hypotheses & Type of Test

p1= the true proportion of low-income preschool children who grow up to need social services

p2= the true proportion of low-income non-preschool children who grow up to need social services

H0: p1 = p2

Ha: p1 < p2

Left-tailed two-proportion z-test α=.05

ConditionsCondition Check

Random samples Assume

Each n < 10% N Assume 62 is < 10% of low-income kids who attend preschool and 61 is < 10% of

low-income kids who don’t

Independent groups Assume

n1p1 ≥ 10

n1(1 – p1) ≥ 10

n2p2≥ 10

n2(1 – p2) ≥ 10

62(.613)≥1062(.387)≥10

61(.803)≥1061(.197)≥10

Test Statistic

• This is still a z-test, so we are still finding z• The statistic in this problem is the difference

between the sample proportions• So we have:

• The parameter is zero, since if H0 were true, the difference between p1 and p2 would be zero

1 2 1 2ˆ ˆ( ) ( )standard deviationp p p pz

Standard Deviation Formula

• Since our null hypothesis claims that p1 = p2 we must use the “special case” standard deviation formula for the denominator

1 2

1 1ˆ ˆ(1 )p pn n

Pooled Estimate for p• The pooled estimate combines the two separate

sample proportions into one estimate:

• In this problem:

1 2

1 2

ˆ x xpn n

38 49ˆ .70762 61

p

Now, to find z…

(.613 .803) 0 2.311 1(.707)(.293)62 61

z

.0104p

Conclusion

• Since p < .05, reject H0. There is enough evidence to conclude that low-income children who attend preschool need social services at a lower rate than those who don’t.

• One more thing – does this prove that preschool causes the lower need for social services?

• No – this is not a controlled experiment!

Homework 22-2

• p. 519 #24, 25, 26