Question

Includes / does not include seems / does not seem
-za za +/-za/2
reject / do not reject insufficient / sufficient

cle, a researcher assertsthat here have been changes in spontaneous ou door acti ties such as bike ding, s 1 ming, and ouch ootball n he a cle he cites separate st dies by o s orts associations hat ndicate bike n ng alone is down 31% rom 199 4 to 2003 ccording to the surveys, 729 of n an a 11 year olds rode a bike at least six times in 1994 and only 46% did in 2003. Assume the sample sizes were 2.000 and 2.500, respectively b. Conduct a lest of hypothesis to answer the quesion posed n part ส. Are the results of parts and b conn,dictory? Explain. a. Calculate ฮ 98% confidence iterval to esim ate the proportion of 7. to 11-year-olds who rode their bike at least six times in 2003. alculate 비 98% oorf dence rte valto stirmateth pro ur on 0 7 to 11-ye비-olds ho rode thei b e .1 easl s·times r 2003. Does thiy suggesi thịt í is lausible o beli ve ₩p upo ion' uf 7 to 11-ves' ulds who 'ode lheir bike t essl sax in es n 20 3 the sem -85in 1 947 Round to four decimal places as needed) Does this suggest that it s plausble to beleve the propotion ot -to 11-year-oids who rode their bike at least six imes in 2003 is the same as in 1551 Since the interval Type sn integer oradecimsl b Lat p, ba tha population praportion from 1994 and lat po ba tha popuiatian praportion from 2003 Idantify tha null and atamativa hypatheses Choose tha corract answar belaw plausible that the proporton of 7- to 11-year-olds who rude ther bike at least six times in 2003 is the same as in 1994 Determine the critical valueis) ต OOTrnw lu separalle nswer3 as needed. Round lu two decimal places as needed.) Calculata the tast statistic Round to thraa decimal places as naaded) Reacl a decision the null hypothesis. There is evidence that the proportion of 7 to 11 year olds who rode their bike at least ix times in 2003 is different than in 1994 Ara tha results of pats a and h cantradictory Explain CA. No, because the resuts from both parts indicate that the proportions for the two years are not different OB. No, because the results from both parts indicate that the proportions for the two years are different O C. Yes, because une part indicates that the pruportions for the two years art! different and the other does not

Answer 1

Basic data:

$\hat{p}_1$ = Sample proportion of 7-11 year olds who rode their bikes at least 6 time in 1994 = 0.72 and 1 - $\hat{p}_1$ = 0.28

n1 = 2000, x1 = 2000 * 0.72 = 1440

$\hat{p}_2$ = Sample proportion of 7-11 year olds who rode their bikes at least 6 time in 2003 = 0.45, and 1 - $\hat{p}_2$ = 0.55

n2 = 2500, x2 = 2500 * 0.45 = 1125

$\hat{p}$ = Overall Proportion = (1440 + 1125) / (2000 + 2500) = 2565//4500 = 0.57

1 - $\hat{p}$ = 0.43

$\alpha$ = 0.02

Zcritical value (2 tailed) = -2.33and +2.333

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(a) 98% Confidence Interval

The Confidence Interval is given by (
P1
- $\hat{p_2}$ ) $\pm$ ME, where

(
P1
- $\hat{p_2}$ ) = 0.72 – 0.45 = 0.27

PMI-P1) . P2(1-p2) 0.72 0.28 0.45 0.55 NIE Zcritical* 2.33* 1 12 2500 0,0329

The Lower Limit = 0.27 - 0.0329 = 0.2371

The Upper Limit = 0.27 + 0.0329 = 0.3029

The 98% Confidence Interval is 0.2371 < p1 - p2 < 0.3029

Since the interval (0.2371,0.3029) does not contain 0, it is not plausible that the proportion of 7 to 11 year olds who rode their bikes 6 times in 2003 is the same as in 1994.

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(b) 2 tailed test for difference in proportions

The Hypothesis: Option C

H0: p1 - p2 = 0

Ha: p1 - p2 $\neq$ 0

The critical values are Z = -2.33,2.33

The Test Statistic:

pl - p2 0.72-0.45 = 19.149 2000 2500

Z = 19.149

Reject the Null Hypothesis. There is sufficient evidence that the proportion of 7 to 11 year olds who rode their bikes at least 6 times in 2003 is different from that in 1994.

Are the results contradictory: Option B: No, because the results from both parts indicate that the proportions for the two years are different.