classtype
small
regular
regular
small
small
regular
small
small
regular
regular
regular
regular
regular
regular
small
regular
small
regular
small
small
small
small
regular
small
small
regular
small
small
regular
small
regular
small
regular
small
regular
small
regular
small
regular
regular
reading
math
578
612
583
661
614
610
595
665
616
624
593
599
693
545
565
654
686
570
529
582
608
572
562
641
644
565
650
552
636
567
554
595
620
639
633
614
606
578
536
629
graduated
610
612
606
648
636
603
610
631
636
626
601
564
752
565
576
684
663
601
583
582
583
562
593
622
639
594
655
594
659
556
577
636
678
655
684
593
599
603
558
639
1
1
1
1
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
0
0
1
1
1
1
0
0
0
1
1
0
1
1
0
1
1
1
1
1
1
regular
regular
small
small
small
small
small
regular
small
small
small
regular
regular
regular
small
small
regular
regular
small
regular
small
small
small
small
regular
regular
small
small
small
small
small
small
regular
regular
regular
small
small
small
regular
small
regular
675
604
577
713
693
595
593
599
616
620
592
702
636
654
675
665
573
641
533
616
670
654
686
693
614
606
614
616
616
661
626
713
636
675
629
680
629
631
650
654
618
655
618
567
672
726
610
559
624
614
596
629
711
608
651
678
639
561
642
568
606
711
639
700
691
624
580
648
711
663
620
672
691
631
700
622
691
651
691
648
691
597
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
regular
small
small
regular
regular
regular
regular
small
small
regular
small
regular
regular
regular
regular
small
small
small
regular
regular
small
small
small
regular
small
small
small
regular
regular
regular
regular
regular
small
small
regular
small
regular
regular
regular
small
regular
604
657
675
665
657
569
650
600
675
665
665
602
618
608
647
606
592
639
641
657
633
650
713
614
588
713
647
657
585
654
552
661
650
686
702
713
549
597
590
647
728
636
636
672
645
663
596
596
616
684
639
659
626
620
648
651
620
585
610
601
752
603
678
726
629
593
684
645
648
620
667
616
648
648
667
678
700
559
636
622
629
667
1
0
1
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
regular
regular
regular
small
small
small
small
small
regular
small
small
regular
regular
small
regular
small
small
small
small
regular
regular
regular
regular
regular
regular
regular
regular
regular
small
regular
small
regular
regular
regular
small
regular
small
regular
regular
regular
regular
629
680
657
665
665
608
583
629
670
665
665
629
614
608
675
644
650
654
654
595
647
543
713
624
670
680
572
641
654
582
597
633
624
728
675
578
639
680
595
670
569
648
645
631
651
636
645
596
645
684
597
691
667
614
636
678
624
659
684
655
594
672
576
659
612
700
691
593
659
667
629
655
631
614
726
634
585
626
678
629
639
612
1
0
1
1
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
1
1
1
regular
small
small
regular
regular
small
small
regular
regular
regular
regular
small
regular
small
small
small
regular
regular
regular
regular
small
small
small
regular
regular
small
small
small
regular
regular
small
regular
small
small
regular
regular
small
regular
regular
small
regular
602
661
650
661
650
633
590
661
686
580
620
670
614
580
606
614
622
639
633
583
657
728
657
636
606
680
650
590
549
600
580
654
702
670
608
680
636
641
608
580
686
594
626
667
678
645
636
620
636
700
593
672
659
596
589
639
620
610
612
663
591
663
672
642
614
604
667
614
573
571
622
608
691
774
726
558
655
601
588
589
629
691
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
regular
regular
regular
regular
regular
small
small
small
regular
small
regular
regular
small
regular
small
small
regular
small
regular
small
small
small
small
regular
regular
small
regular
regular
small
regular
regular
small
regular
small
regular
regular
regular
small
regular
small
small
622
633
610
593
670
614
599
633
686
702
588
622
693
631
680
657
629
599
665
595
633
661
622
614
680
626
602
557
650
608
631
629
657
602
595
612
702
644
608
636
675
691
629
667
631
629
667
594
629
678
711
634
639
700
593
691
678
601
620
645
626
684
726
620
604
678
603
596
591
626
573
639
604
655
631
614
588
711
648
612
663
651
0
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
small
small
regular
small
small
regular
regular
regular
small
small
small
regular
regular
small
regular
small
small
small
regular
small
regular
small
small
small
regular
regular
regular
small
regular
small
regular
small
regular
small
regular
small
small
small
small
small
regular
633
587
624
597
650
595
670
686
670
595
572
626
631
675
675
636
600
657
588
641
629
612
661
654
602
595
620
639
644
612
665
728
629
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554
650
629
693
654
567
602
620
645
612
634
655
599
655
667
651
616
588
612
558
645
700
601
610
631
608
651
612
620
624
634
567
622
596
599
678
634
667
726
614
691
541
655
636
672
726
558
639
1
1
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
small
small
regular
small
small
small
small
small
regular
small
regular
small
regular
small
small
small
small
regular
small
regular
small
small
small
regular
regular
regular
regular
regular
small
small
regular
small
regular
small
small
regular
regular
regular
small
regular
regular
657
650
557
647
629
569
650
624
641
620
578
644
606
661
602
614
602
626
587
604
629
693
600
629
622
631
593
693
633
647
629
631
629
670
570
616
657
593
657
713
602
651
631
570
684
631
559
659
608
659
639
612
651
582
636
571
604
559
593
593
606
634
667
576
608
648
667
596
651
672
597
594
634
678
642
616
631
614
629
684
663
610
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
0
small
regular
small
regular
regular
small
regular
small
regular
small
small
regular
regular
small
small
regular
small
small
small
small
regular
small
regular
small
small
small
small
regular
small
small
regular
regular
regular
regular
small
regular
regular
regular
small
regular
regular
728
631
633
644
582
661
606
582
608
620
675
597
622
618
644
599
620
629
578
702
686
639
654
626
557
702
670
590
529
587
560
650
604
585
629
647
626
636
686
657
588
684
636
610
629
618
667
667
606
629
604
672
645
618
642
631
614
599
631
601
651
648
636
659
596
564
672
659
622
515
594
544
667
604
591
639
651
599
645
659
634
564
1
1
1
1
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1
0
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1
1
1
1
1
1
1
1
0
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1
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1
1
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1
1
1
1
1
1
1
1
1
0
1
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regular
small
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small
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small
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small
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small
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small
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regular
regular
regular
small
regular
regular
small
small
regular
regular
regular
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665
670
686
661
675
657
587
588
654
686
654
654
608
583
702
626
608
665
593
608
644
639
585
573
657
641
631
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554
650
582
636
657
647
686
753
650
633
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595
629
645
684
651
631
648
672
562
645
659
700
631
636
634
577
678
659
629
678
562
672
634
672
571
568
645
622
672
629
580
667
574
648
684
624
663
678
639
624
577
583
645
1
1
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549
614
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622
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644
580
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587
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602
570
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670
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610
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728
631
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650
547
603
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624
589
636
601
672
642
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601
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610
659
648
616
585
752
639
591
663
648
629
624
639
659
616
642
574
700
634
691
579
634
604
599
606
691
663
711
678
0
1
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633
654
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562
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564
614
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573
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580
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557
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545
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680
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675
633
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651
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616
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579
614
586
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588
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659
636
574
616
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645
556
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608
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645
1
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small
small
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small
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629
610
657
604
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577
644
567
636
618
624
597
600
624
693
606
675
647
616
578
616
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753
636
555
614
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585
588
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610
648
588
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624
612
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576
610
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614
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678
626
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594
672
606
585
589
659
626
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655
645
620
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629
700
586
593
631
586
604
596
589
1
1
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636
622
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650
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592
559
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585
626
663
711
624
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651
580
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1
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670
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728
641
592
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654
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1
——————————————————————————-Name
Description
——————– ———————————————————`classtype` Type of kindergarten class (small, regular)
`reading`
Total scaled score for reading portion of fourth grade standardized test
`graduated` High school graduation (did graduate = 1, did not graduate = 0)
——————————————————————————-? Please download and load STAR.csv Download STAR.csv in R.
? We are interested in the average treatment effect of a small class on students’
performance. This is your “parameter.”
? Q1. Then, what is the null hypothesis? What is the alternative hypothesis?
? Q2. Estimate the average treatment effect of a small class on reading scores
and save the value to “Estimated_reading”. Since it is a randomized experiment,
you use a difference-in-means estimator. (Please refer to pset 3 for computing
this estimate.)
? Q3. In this research, what is the estimate? what is the estimator?
? Q4. Now you are calculating a test statistic (Z-score). If the standard error is
2.06, what is the test statistic? Save this value to “test_stat” and report it. (Hint:
recall how we compute the z-score when the null hypothesis is assumed to be
true. Go back to the lecture slide.)
? Q5. Based on the result of a test statistic, can you reject the null hypothesis at
the 5% significance level? Why? Why not? (Hint: what is the critical value at the
5% significance level)
? Q6. Let’s also calculate the p-value. To compute p-values in R, we can use the
function pnorm() in conjunction with the function abs(), which stands for
bsolute value. Please run 2*pnorm(-abs(test_stat)). Is the p-value less than
0.05? Is this result evidence of “rejecting the null hypothesis”?

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