Worked solution P65-Easy-1. The pooled estimate is 0.5522, z=1.662, and the two-sided p-value is 0.0965. $${\widehat{p}}_c=\frac{67+60}{110+120}=0.5522,z=\frac{0.6091-0.5000}{0.0656}=1.662.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-2. The pooled estimate is 0.4914, z=1.697, and the two-sided p-value is 0.0896. $${\widehat{p}}_c=\frac{61+53}{111+121}=0.4914,z=\frac{0.5495-0.4380}{0.0657}=1.697.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-3. The pooled estimate is 0.5214, z=1.731, and the two-sided p-value is 0.0835. $${\widehat{p}}_c=\frac{65+57}{112+122}=0.5214,z=\frac{0.5804-0.4672}{0.0654}=1.731.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-4. The pooled estimate is 0.5127, z=1.581, and the two-sided p-value is 0.1139. $${\widehat{p}}_c=\frac{64+57}{113+123}=0.5127,z=\frac{0.5664-0.4634}{0.0651}=1.581.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-5. The pooled estimate is 0.5546, z=1.768, and the two-sided p-value is 0.0770. $${\widehat{p}}_c=\frac{70+62}{114+124}=0.5546,z=\frac{0.6140-0.5000}{0.0645}=1.768.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-6. The pooled estimate is 0.5333, z=1.727, and the two-sided p-value is 0.0842. $${\widehat{p}}_c=\frac{68+60}{115+125}=0.5333,z=\frac{0.5913-0.4800}{0.0645}=1.727.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-7. The pooled estimate is 0.5537, z=1.752, and the two-sided p-value is 0.0798. $${\widehat{p}}_c=\frac{71+63}{116+126}=0.5537,z=\frac{0.6121-0.5000}{0.0640}=1.752.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-8. The pooled estimate is 0.5410, z=1.724, and the two-sided p-value is 0.0847. $${\widehat{p}}_c=\frac{70+62}{117+127}=0.5410,z=\frac{0.5983-0.4882}{0.0639}=1.724.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-9. The pooled estimate is 0.5041, z=1.664, and the two-sided p-value is 0.0960. $${\widehat{p}}_c=\frac{66+58}{118+128}=0.5041,z=\frac{0.5593-0.4531}{0.0638}=1.664.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-10. The pooled estimate is 0.5242, z=1.685, and the two-sided p-value is 0.0920. $${\widehat{p}}_c=\frac{69+61}{119+129}=0.5242,z=\frac{0.5798-0.4729}{0.0635}=1.685.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-11. The pooled estimate is 0.4920, z=1.762, and the two-sided p-value is 0.0780. $${\widehat{p}}_c=\frac{66+57}{120+130}=0.4920,z=\frac{0.5500-0.4385}{0.0633}=1.762.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-12. The pooled estimate is 0.5119, z=1.781, and the two-sided p-value is 0.0750. $${\widehat{p}}_c=\frac{69+60}{121+131}=0.5119,z=\frac{0.5702-0.4580}{0.0630}=1.781.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-13. The pooled estimate is 0.5630, z=1.852, and the two-sided p-value is 0.0640. $${\widehat{p}}_c=\frac{76+67}{122+132}=0.5630,z=\frac{0.6230-0.5076}{0.0623}=1.852.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-14. The pooled estimate is 0.5625, z=1.718, and the two-sided p-value is 0.0858. $${\widehat{p}}_c=\frac{76+68}{123+133}=0.5625,z=\frac{0.6179-0.5113}{0.0621}=1.718.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-15. The pooled estimate is 0.5426, z=1.679, and the two-sided p-value is 0.0931. $${\widehat{p}}_c=\frac{74+66}{124+134}=0.5426,z=\frac{0.5968-0.4925}{0.0621}=1.679.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Easy-16. The pooled estimate is 0.5115, z=1.753, and the two-sided p-value is 0.0797. $${\widehat{p}}_c=\frac{71+62}{125+135}=0.5115,z=\frac{0.5680-0.4593}{0.0620}=1.753.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-1. The pooled estimate is 0.5130, z=1.734, and the two-sided p-value is 0.0830. $${\widehat{p}}_c=\frac{63+55}{110+120}=0.5130,z=\frac{0.5727-0.4583}{0.0660}=1.734.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-2. The pooled estimate is 0.4914, z=1.697, and the two-sided p-value is 0.0896. $${\widehat{p}}_c=\frac{61+53}{111+121}=0.4914,z=\frac{0.5495-0.4380}{0.0657}=1.697.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-3. The pooled estimate is 0.4957, z=1.696, and the two-sided p-value is 0.0899. $${\widehat{p}}_c=\frac{62+54}{112+122}=0.4957,z=\frac{0.5536-0.4426}{0.0654}=1.696.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-4. The pooled estimate is 0.4915, z=1.683, and the two-sided p-value is 0.0923. $${\widehat{p}}_c=\frac{62+54}{113+123}=0.4915,z=\frac{0.5487-0.4390}{0.0651}=1.683.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-5. The pooled estimate is 0.5126, z=1.704, and the two-sided p-value is 0.0884. $${\widehat{p}}_c=\frac{65+57}{114+124}=0.5126,z=\frac{0.5702-0.4597}{0.0649}=1.704.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-6. The pooled estimate is 0.5167, z=1.702, and the two-sided p-value is 0.0887. $${\widehat{p}}_c=\frac{66+58}{115+125}=0.5167,z=\frac{0.5739-0.4640}{0.0646}=1.702.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-7. The pooled estimate is 0.5537, z=1.752, and the two-sided p-value is 0.0798. $${\widehat{p}}_c=\frac{71+63}{116+126}=0.5537,z=\frac{0.6121-0.5000}{0.0640}=1.752.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-8. The pooled estimate is 0.5410, z=1.724, and the two-sided p-value is 0.0847. $${\widehat{p}}_c=\frac{70+62}{117+127}=0.5410,z=\frac{0.5983-0.4882}{0.0639}=1.724.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-9. The pooled estimate is 0.5041, z=1.664, and the two-sided p-value is 0.0960. $${\widehat{p}}_c=\frac{66+58}{118+128}=0.5041,z=\frac{0.5593-0.4531}{0.0638}=1.664.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-10. The pooled estimate is 0.5040, z=1.785, and the two-sided p-value is 0.0743. $${\widehat{p}}_c=\frac{67+58}{119+129}=0.5040,z=\frac{0.5630-0.4496}{0.0635}=1.785.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-11. The pooled estimate is 0.4920, z=1.762, and the two-sided p-value is 0.0780. $${\widehat{p}}_c=\frac{66+57}{120+130}=0.4920,z=\frac{0.5500-0.4385}{0.0633}=1.762.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-12. The pooled estimate is 0.5317, z=1.683, and the two-sided p-value is 0.0925. $${\widehat{p}}_c=\frac{71+63}{121+131}=0.5317,z=\frac{0.5868-0.4809}{0.0629}=1.683.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-13. The pooled estimate is 0.5512, z=1.706, and the two-sided p-value is 0.0880. $${\widehat{p}}_c=\frac{74+66}{122+132}=0.5512,z=\frac{0.6066-0.5000}{0.0625}=1.706.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-14. The pooled estimate is 0.5117, z=1.766, and the two-sided p-value is 0.0773. $${\widehat{p}}_c=\frac{70+61}{123+133}=0.5117,z=\frac{0.5691-0.4586}{0.0625}=1.766.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-15. The pooled estimate is 0.5543, z=1.823, and the two-sided p-value is 0.0683. $${\widehat{p}}_c=\frac{76+67}{124+134}=0.5543,z=\frac{0.6129-0.5000}{0.0619}=1.823.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Tough-16. The pooled estimate is 0.5231, z=1.893, and the two-sided p-value is 0.0584. $${\widehat{p}}_c=\frac{73+63}{125+135}=0.5231,z=\frac{0.5840-0.4667}{0.0620}=1.893.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-1. The pooled estimate is 0.5130, z=1.734, and the two-sided p-value is 0.0830. $${\widehat{p}}_c=\frac{63+55}{110+120}=0.5130,z=\frac{0.5727-0.4583}{0.0660}=1.734.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-2. The pooled estimate is 0.5129, z=1.595, and the two-sided p-value is 0.1108. $${\widehat{p}}_c=\frac{63+56}{111+121}=0.5129,z=\frac{0.5676-0.4628}{0.0657}=1.595.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-3. The pooled estimate is 0.5043, z=1.707, and the two-sided p-value is 0.0878. $${\widehat{p}}_c=\frac{63+55}{112+122}=0.5043,z=\frac{0.5625-0.4508}{0.0654}=1.707.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-4. The pooled estimate is 0.5127, z=1.581, and the two-sided p-value is 0.1139. $${\widehat{p}}_c=\frac{64+57}{113+123}=0.5127,z=\frac{0.5664-0.4634}{0.0651}=1.581.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-5. The pooled estimate is 0.5126, z=1.704, and the two-sided p-value is 0.0884. $${\widehat{p}}_c=\frac{65+57}{114+124}=0.5126,z=\frac{0.5702-0.4597}{0.0649}=1.704.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-6. The pooled estimate is 0.5417, z=1.740, and the two-sided p-value is 0.0819. $${\widehat{p}}_c=\frac{69+61}{115+125}=0.5417,z=\frac{0.6000-0.4880}{0.0644}=1.740.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-7. The pooled estimate is 0.5124, z=1.689, and the two-sided p-value is 0.0912. $${\widehat{p}}_c=\frac{66+58}{116+126}=0.5124,z=\frac{0.5690-0.4603}{0.0643}=1.689.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-8. The pooled estimate is 0.5246, z=1.699, and the two-sided p-value is 0.0892. $${\widehat{p}}_c=\frac{68+60}{117+127}=0.5246,z=\frac{0.5812-0.4724}{0.0640}=1.699.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-9. The pooled estimate is 0.5610, z=1.750, and the two-sided p-value is 0.0801. $${\widehat{p}}_c=\frac{73+65}{118+128}=0.5610,z=\frac{0.6186-0.5078}{0.0633}=1.750.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-10. The pooled estimate is 0.5323, z=1.697, and the two-sided p-value is 0.0897. $${\widehat{p}}_c=\frac{70+62}{119+129}=0.5323,z=\frac{0.5882-0.4806}{0.0634}=1.697.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-11. The pooled estimate is 0.5000, z=1.772, and the two-sided p-value is 0.0763. $${\widehat{p}}_c=\frac{67+58}{120+130}=0.5000,z=\frac{0.5583-0.4462}{0.0633}=1.772.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-12. The pooled estimate is 0.5040, z=1.770, and the two-sided p-value is 0.0767. $${\widehat{p}}_c=\frac{68+59}{121+131}=0.5040,z=\frac{0.5620-0.4504}{0.0630}=1.770.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-13. The pooled estimate is 0.5315, z=1.801, and the two-sided p-value is 0.0716. $${\widehat{p}}_c=\frac{72+63}{122+132}=0.5315,z=\frac{0.5902-0.4773}{0.0627}=1.801.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-14. The pooled estimate is 0.5234, z=1.657, and the two-sided p-value is 0.0974. $${\widehat{p}}_c=\frac{71+63}{123+133}=0.5234,z=\frac{0.5772-0.4737}{0.0625}=1.657.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-15. The pooled estimate is 0.5000, z=1.745, and the two-sided p-value is 0.0811. $${\widehat{p}}_c=\frac{69+60}{124+134}=0.5000,z=\frac{0.5565-0.4478}{0.0623}=1.745.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.

Worked solution P65-Toughest-16. The pooled estimate is 0.5538, z=1.690, and the two-sided p-value is 0.0910. $${\widehat{p}}_c=\frac{76+68}{125+135}=0.5538,z=\frac{0.6080-0.5037}{0.0617}=1.690.$$ Interpretation: Compare the p-value with the stated alpha and make a population conclusion in the direction specified by Ha. Validity: Require independent random groups or randomized treatments, within-group independence, and pooled expected successes and failures of at least 10 in each group. Error to reject: The pooled proportion is justified only under the null equality; do not use it for the matching confidence interval.