One sample t test for the mean | Tests and confidence intervals Dynamic

Upper-tail probability $p$
df 0.10.050.0250.010.005
13.0786.31412.70631.82163.657
21.8862.924.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.441.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.862.3062.8963.355
91.3831.8332.2622.8213.25
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.351.7712.162.653.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947
161.3371.7462.122.5832.921
171.3331.742.112.5672.898
181.331.7342.1012.5522.878
191.3281.7292.0932.5392.861
201.3251.7252.0862.5282.845
211.3231.7212.082.5182.831
221.3211.7172.0742.5082.819
231.3191.7142.0692.52.807
241.3181.7112.0642.4922.797
251.3161.7082.062.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.311.6972.0422.4572.75
401.3031.6842.0212.4232.704
501.2991.6762.0092.4032.678
601.2961.67122.392.66
1001.291.661.9842.3642.626

Upper-tail probability $p$
df 0.250.20.150.10.050.0250.020.010.0050.00250.0010.0005
11.321.642.072.713.845.025.416.637.889.1410.8312.12
22.773.223.794.615.997.387.829.2110.611.9813.8215.2
34.114.645.326.257.819.359.8411.3412.8414.3216.2717.73
45.395.996.747.789.4911.1411.6713.2814.8616.4218.4720
56.637.298.129.2411.0712.8313.3915.0916.7518.3920.5222.11
67.848.569.4510.6412.5914.4515.0316.8118.5520.2522.4624.1
79.049.810.7512.0214.0716.0116.6218.4820.2822.0424.3226.02
810.2211.0312.0313.3615.5117.5318.1720.0921.9523.7726.1227.87
911.3912.2413.2914.6816.9219.0219.6821.6723.5925.4627.8829.67
1012.5513.4414.5315.9918.3120.4821.1623.2125.1927.1129.5931.42
1113.714.6315.7717.2819.6821.9222.6224.7226.7628.7331.2633.14
1214.8515.8116.9918.5521.0323.3424.0526.2228.330.3232.9134.82
1315.9816.9818.219.8122.3624.7425.4727.6929.8231.8834.5336.48
1417.1218.1519.4121.0623.6826.1226.8729.1431.3233.4336.1238.11
1518.2519.3120.622.312527.4928.2630.5832.834.9537.739.72
1619.3720.4721.7923.5426.328.8529.633234.2736.4639.2541.31
1720.4921.6122.9824.7727.5930.193133.4135.7237.9540.7942.88
1821.622.7624.1625.9928.8731.5332.3534.8137.1639.4242.3144.43
1922.7223.925.3327.230.1432.8533.6936.1938.5840.8843.8245.97
2023.8325.0426.528.4131.4134.1735.0237.574042.3445.3147.5
2124.9326.1727.6629.6232.6735.4836.3438.9341.443.7846.849.01
2226.0427.328.8230.8133.9236.7837.6640.2942.845.248.2750.51
2327.1428.4329.9832.0135.1738.0838.9741.6444.1846.6249.7352
2428.2429.5531.1333.236.4239.3640.2742.9845.5648.0351.1853.48
2529.3430.6832.2834.3837.6540.6541.5744.3146.9349.4452.6254.95
2630.4331.7933.4335.5638.8941.9242.8645.6448.2950.8354.0556.41
2731.5332.9134.5736.7440.1143.1944.1446.9649.6452.2255.4857.86
2832.6234.0335.7137.9241.3444.4645.4248.2850.9953.5956.8959.3
2933.7135.1436.8539.0942.5645.7246.6949.5952.3454.9758.360.73
3034.836.2537.9940.2643.7746.9847.9650.8953.6756.3359.762.16
4045.6247.2749.2451.8155.7659.3460.4463.6966.7769.773.476.09
5056.3358.1660.3563.1767.571.4272.6176.1579.4982.6686.6689.56
6066.9868.9771.3474.479.0883.384.5888.3891.9595.3499.61102.69
7077.5879.7182.2685.5390.5395.0296.39100.43104.21107.81112.32115.58
8088.1390.4193.1196.58101.88106.63108.07112.33116.32120.1124.84128.26
100109.14111.67114.66118.5124.34129.56131.14135.81140.17144.29149.45153.17

df numerator
df denominator p 1 2 3 4 5 6 7
180.13.012.622.422.292.22.132.08
180.054.413.553.162.932.772.662.58
180.0255.984.563.953.613.383.223.1
180.018.296.015.094.584.254.013.84
180.00115.3810.398.497.466.816.356.02
190.12.992.612.42.272.182.112.06
190.054.383.523.132.92.742.632.54
190.0255.924.513.93.563.333.173.05
190.018.185.935.014.54.173.943.77
190.00115.0810.168.287.276.626.185.85
200.12.972.592.382.252.162.092.04
200.054.353.493.12.872.712.62.51
200.0255.874.463.863.513.293.133.01
200.018.15.854.944.434.13.873.7
200.00114.829.958.17.16.466.025.69
290.12.892.52.282.152.061.991.93
290.054.183.332.932.72.552.432.35
290.0255.594.23.613.273.042.882.76
290.017.65.424.544.043.733.53.33
290.00113.398.857.126.195.595.184.87
500.12.812.412.22.061.971.91.84
500.054.033.182.792.562.42.292.2
500.0255.343.973.393.052.832.672.55
500.017.175.064.23.723.413.193.02
500.00112.227.966.345.464.94.514.22

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
-3.40.00030.00030.00030.00030.00030.00030.00030.00030.00030.0002
-3.30.00050.00050.00050.00040.00040.00040.00040.00040.00040.0003
-3.20.00070.00070.00060.00060.00060.00060.00060.00050.00050.0005
-3.10.00100.00090.00090.00090.00080.00080.00080.00080.00070.0007
-30.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010
-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014
-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019
-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026
-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036
-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048
-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064
-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084
-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110
-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143
-20.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183
-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233
-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294
-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367
-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455
-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559
-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681
-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823
-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985
-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170
-10.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379
-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611
-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867
-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148
-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451
-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776
-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121
-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483
-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859
-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247
-00.50000.49600.49200.48800.48400.48010.47610.47210.46810.4641
00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389
10.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
20.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
30.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
3.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993
3.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995
3.30.99950.99950.99950.99960.99960.99960.99960.99960.99960.9997
3.40.99970.99970.99970.99970.99970.99970.99970.99970.99970.9998

Table entries are the probability that the number of successes in n trials is x, based on the binomial distribution with success probability P

P
n x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
100.90000.80000.70000.60000.50000.40000.30000.20000.1000
10.10000.20000.30000.40000.50000.60000.70000.80000.9000
200.81000.64000.49000.36000.25000.16000.09000.04000.0100
10.18000.32000.42000.48000.50000.48000.42000.32000.1800
20.01000.04000.09000.16000.25000.36000.49000.64000.8100
300.72900.51200.34300.21600.12500.06400.02700.00800.0010
10.24300.38400.44100.43200.37500.28800.18900.09600.0270
20.02700.09600.18900.28800.37500.43200.44100.38400.2430
30.00100.00800.02700.06400.12500.21600.34300.51200.7290
400.65610.40960.24010.12960.06250.02560.00810.00160.0001
10.29160.40960.41160.34560.25000.15360.07560.02560.0036
20.04860.15360.26460.34560.37500.34560.26460.15360.0486
30.00360.02560.07560.15360.25000.34560.41160.40960.2916
40.00010.00160.00810.02560.06250.12960.24010.40960.6561
500.59050.32770.16810.07780.03130.01020.00240.00030.0000
10.32810.40960.36020.25920.15630.07680.02840.00640.0005
20.07290.20480.30870.34560.31250.23040.13230.05120.0081
30.00810.05120.13230.23040.31250.34560.30870.20480.0729
40.00050.00640.02840.07680.15630.25920.36020.40960.3281
50.00000.00030.00240.01020.03130.07780.16810.32770.5905
600.53140.26210.11760.04670.01560.00410.00070.00010.0000
10.35430.39320.30250.18660.09380.03690.01020.00150.0001
20.09840.24580.32410.31100.23440.13820.05950.01540.0012
30.01460.08190.18520.27650.31250.27650.18520.08190.0146
40.00120.01540.05950.13820.23440.31100.32410.24580.0984
50.00010.00150.01020.03690.09380.18660.30250.39320.3543
60.00000.00010.00070.00410.01560.04670.11760.26210.5314
700.47830.20970.08240.02800.00780.00160.00020.00000.0000
10.37200.36700.24710.13060.05470.01720.00360.00040.0000
20.12400.27530.31770.26130.16410.07740.02500.00430.0002
30.02300.11470.22690.29030.27340.19350.09720.02870.0026
40.00260.02870.09720.19350.27340.29030.22690.11470.0230
50.00020.00430.02500.07740.16410.26130.31770.27530.1240
60.00000.00040.00360.01720.05470.13060.24710.36700.3720
70.00000.00000.00020.00160.00780.02800.08240.20970.4783
800.43050.16780.05760.01680.00390.00070.00010.00000.0000
10.38260.33550.19770.08960.03130.00790.00120.00010.0000
20.14880.29360.29650.20900.10940.04130.01000.00110.0000
30.03310.14680.25410.27870.21880.12390.04670.00920.0004
40.00460.04590.13610.23220.27340.23220.13610.04590.0046
50.00040.00920.04670.12390.21880.27870.25410.14680.0331
60.00000.00110.01000.04130.10940.20900.29650.29360.1488
70.00000.00010.00120.00790.03130.08960.19770.33550.3826
80.00000.00000.00010.00070.00390.01680.05760.16780.4305
900.38740.13420.04040.01010.00200.00030.00000.00000.0000
10.38740.30200.15560.06050.01760.00350.00040.00000.0000
20.17220.30200.26680.16120.07030.02120.00390.00030.0000
30.04460.17620.26680.25080.16410.07430.02100.00280.0001
40.00740.06610.17150.25080.24610.16720.07350.01650.0008
50.00080.01650.07350.16720.24610.25080.17150.06610.0074
60.00010.00280.02100.07430.16410.25080.26680.17620.0446
70.00000.00030.00390.02120.07030.16120.26680.30200.1722
80.00000.00000.00040.00350.01760.06050.15560.30200.3874
90.00000.00000.00000.00030.00200.01010.04040.13420.3874
1000.34870.10740.02820.00600.00100.00010.00000.00000.0000
10.38740.26840.12110.04030.00980.00160.00010.00000.0000
20.19370.30200.23350.12090.04390.01060.00140.00010.0000
30.05740.20130.26680.21500.11720.04250.00900.00080.0000
40.01120.08810.20010.25080.20510.11150.03680.00550.0001
50.00150.02640.10290.20070.24610.20070.10290.02640.0015
60.00010.00550.03680.11150.20510.25080.20010.08810.0112
70.00000.00080.00900.04250.11720.21500.26680.20130.0574
80.00000.00010.00140.01060.04390.12090.23350.30200.1937
90.00000.00000.00010.00160.00980.04030.12110.26840.3874
100.00000.00000.00000.00010.00100.00600.02820.10740.3487
1100.31380.08590.01980.00360.00050.00000.00000.00000.0000
10.38350.23620.09320.02660.00540.00070.00000.00000.0000
20.21310.29530.19980.08870.02690.00520.00050.00000.0000
30.07100.22150.25680.17740.08060.02340.00370.00020.0000
40.01580.11070.22010.23650.16110.07010.01730.00170.0000
50.00250.03880.13210.22070.22560.14710.05660.00970.0003
60.00030.00970.05660.14710.22560.22070.13210.03880.0025
70.00000.00170.01730.07010.16110.23650.22010.11070.0158
80.00000.00020.00370.02340.08060.17740.25680.22150.0710
90.00000.00000.00050.00520.02690.08870.19980.29530.2131
100.00000.00000.00000.00070.00540.02660.09320.23620.3835
110.00000.00000.00000.00000.00050.00360.01980.08590.3138
1200.28240.06870.01380.00220.00020.00000.00000.00000.0000
10.37660.20620.07120.01740.00290.00030.00000.00000.0000
20.23010.28350.16780.06390.01610.00250.00020.00000.0000
30.08520.23620.23970.14190.05370.01250.00150.00010.0000
40.02130.13290.23110.21280.12080.04200.00780.00050.0000
50.00380.05320.15850.22700.19340.10090.02910.00330.0000
60.00050.01550.07920.17660.22560.17660.07920.01550.0005
70.00000.00330.02910.10090.19340.22700.15850.05320.0038
80.00000.00050.00780.04200.12080.21280.23110.13290.0213
90.00000.00010.00150.01250.05370.14190.23970.23620.0852
100.00000.00000.00020.00250.01610.06390.16780.28350.2301
110.00000.00000.00000.00030.00290.01740.07120.20620.3766
120.00000.00000.00000.00000.00020.00220.01380.06870.2824
1300.25420.05500.00970.00130.00010.00000.00000.00000.0000
10.36720.17870.05400.01130.00160.00010.00000.00000.0000
20.24480.26800.13880.04530.00950.00120.00010.00000.0000
30.09970.24570.21810.11070.03490.00650.00060.00000.0000
40.02770.15350.23370.18450.08730.02430.00340.00010.0000
50.00550.06910.18030.22140.15710.06560.01420.00110.0000
60.00080.02300.10300.19680.20950.13120.04420.00580.0001
70.00010.00580.04420.13120.20950.19680.10300.02300.0008
80.00000.00110.01420.06560.15710.22140.18030.06910.0055
90.00000.00010.00340.02430.08730.18450.23370.15350.0277
100.00000.00000.00060.00650.03490.11070.21810.24570.0997
110.00000.00000.00010.00120.00950.04530.13880.26800.2448
120.00000.00000.00000.00010.00160.01130.05400.17870.3672
130.00000.00000.00000.00000.00010.00130.00970.05500.2542
1400.22880.04400.00680.00080.00010.00000.00000.00000.0000
10.35590.15390.04070.00730.00090.00010.00000.00000.0000
20.25700.25010.11340.03170.00560.00050.00000.00000.0000
30.11420.25010.19430.08450.02220.00330.00020.00000.0000
40.03490.17200.22900.15490.06110.01360.00140.00000.0000
50.00780.08600.19630.20660.12220.04080.00660.00030.0000
60.00130.03220.12620.20660.18330.09180.02320.00200.0000
70.00020.00920.06180.15740.20950.15740.06180.00920.0002
80.00000.00200.02320.09180.18330.20660.12620.03220.0013
90.00000.00030.00660.04080.12220.20660.19630.08600.0078
100.00000.00000.00140.01360.06110.15490.22900.17200.0349
110.00000.00000.00020.00330.02220.08450.19430.25010.1142
120.00000.00000.00000.00050.00560.03170.11340.25010.2570
130.00000.00000.00000.00010.00090.00730.04070.15390.3559
140.00000.00000.00000.00000.00010.00080.00680.04400.2288
1500.20590.03520.00470.00050.00000.00000.00000.00000.0000
10.34320.13190.03050.00470.00050.00000.00000.00000.0000
20.26690.23090.09160.02190.00320.00030.00000.00000.0000
30.12850.25010.17000.06340.01390.00160.00010.00000.0000
40.04280.18760.21860.12680.04170.00740.00060.00000.0000
50.01050.10320.20610.18590.09160.02450.00300.00010.0000
60.00190.04300.14720.20660.15270.06120.01160.00070.0000
70.00030.01380.08110.17710.19640.11810.03480.00350.0000
80.00000.00350.03480.11810.19640.17710.08110.01380.0003
90.00000.00070.01160.06120.15270.20660.14720.04300.0019
100.00000.00010.00300.02450.09160.18590.20610.10320.0105
110.00000.00000.00060.00740.04170.12680.21860.18760.0428
120.00000.00000.00010.00160.01390.06340.17000.25010.1285
130.00000.00000.00000.00030.00320.02190.09160.23090.2669
140.00000.00000.00000.00000.00050.00470.03050.13190.3432
150.00000.00000.00000.00000.00000.00050.00470.03520.2059

A researcher is interested in the level of life satisfaction of Dutch women. In a random sample of $22$ Dutch women, she finds an average life satisfaction score of $\bar{y} = 68.4$ (on a scale from 0 to 100). The corresponding $98$% confidence interval for the population mean ranges from $63.03$ to $ 73.77$.

Given the confidence interval, how large is the margin of error ($m$)? And how large is the standard error of the sample mean ($SE$)?

$ m$ = 7.16
$ SE$ = 2.454
$ m$ = 7.16
$ SE$ = 3.089
$ m$ = 5.37
$ SE$ = 1.976
$ m$ = 5.37
$ SE$ = 2.133
I don't know...

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Margin of error

Margin of error

I will show you how to find the margin of error, given the confidence interval. As I said in the previous section, the $C$% confidence interval for the population mean $\mu$ is computed as: $$CI = \bar{y} \pm t^* \times \frac{s}{\sqrt{N}}$$ The term $t^* \times \frac{s}{\sqrt{N}}$ is the margin of error. Hence, the confidence interval ranges from $\bar{y}$ minus the margin of error, to $\bar{y}$ plus the margin of error:

Confidence interval and margin of error
This means that the width of the confidence interval is twice the margin of error. Hence, the margin of error is half the width of the confidence interval.

The width of the confidence interval found by the researcher is $73.77 - 63.03 = 10.74$. Hence, the margin of error is equal to $10.74 / 2 = 5.37$.

Standard error

Your answer regarding the margin of error is correct! However, your answer regarding the standard error is incorrect. Let me give you a hint. The $C$% confidence interval for the population mean $\mu$ is computed as: $$CI = \bar{y} \pm t^* \times \frac{s}{\sqrt{N}}$$ where $\bar{y}$ is the sample mean, $s$ is the sample standard deviation, $N$ is the sample size, and $t^*$ is the value under the $t_{N-1}$ distribution with area $C/100$ between $-t^*$ and $t^*$.

The term $t^* \times \frac{s}{\sqrt{N}}$ is the margin of error $m$: $$m = t^* \times \frac{s}{\sqrt{N}}$$ The term $\frac{s}{\sqrt{N}}$ is the standard error of the sample mean. Hence, the margin of error $m$ is equal to $t^*$ times the standard error $\frac{s}{\sqrt{N}}$: $$m = t^* \times SE$$ This means that you can compute the standard error $SE$ if you know $t^*$ and the margin of error $m$. Can you do it?

Standard error

Let me show you how to find the standard error, given the confidence interval. The $C$% confidence interval for the population mean $\mu$ is computed as: $$CI = \bar{y} \pm t^* \times \frac{s}{\sqrt{N}}$$ where $\bar{y}$ is the sample mean, $s$ is the sample standard deviation, $N$ is the sample size, and $t^*$ is the value under the $t_{N-1}$ distribution with area $C/100$ between $-t^*$ and $t^*$.

The term $t^* \times \frac{s}{\sqrt{N}}$ is the margin of error $m$: $$m = t^* \times \frac{s}{\sqrt{N}}$$ The term $\frac{s}{\sqrt{N}}$ is the standard error of the sample mean. Hence, the margin of error $m$ is equal to $t^*$ times the standard error $\frac{s}{\sqrt{N}}$: $$m = t^* \times SE$$ This means that we can compute the standard error $SE$ if we know $t^*$ and the margin of error $m$, as: $$SE = \frac{m}{t^*}$$ The value for $t^*$ you can find in the table with critical $t$ values. You need the value under the $t_{N-1}$ distribution with area $0.98$ between $-t^*$ and $t^*$. This is the value in the column for upper tail probability $\frac{1 - 0.98}{2} = 0.01$ and the row for $N - 1 = 22 - 1 = 21$ degrees of freedom. You will find a $t^*$ of $2.518$. The margin of error $m$ is equal to half the width of the confidence interval, which is $m = (73.77 - 63.03) / 2 = 5.37$ (see the explanation for the margin of error). Now that we know $t^*$ and the margin of error $m$, we can find the standard error as $$ \begin{aligned} SE &= \frac{m}{t^*}\\ &= \frac{5.37}{2.518}\\ &= 2.133 \end{aligned} $$

Margin of error and standard error

Let me help you start. The $C$% confidence interval for the population mean $\mu$ is computed as: $$CI = \bar{y} \pm t^* \times \frac{s}{\sqrt{N}}$$ where $\bar{y}$ is the sample mean, $s$ is the sample standard deviation, $N$ is the sample size, and $t^*$ is the value under the $t_{N-1}$ distribution with area $C/100$ between $-t^*$ and $t^*$.


The term $t^* \times \frac{s}{\sqrt{N}}$ is the margin of error. Hence, the confidence interval ranges from $\bar{y}$ minus the margin of error, to $\bar{y}$ plus the margin of error:

Confidence interval and margin of error
Given that the confidence interval ranges from $63.03$ to $73.77$, how large must be the margin of error?


The term $\frac{s}{\sqrt{N}}$ is the standard error of the sample mean. Hence, the margin of error $m$ is equal to $t^*$ times the standard error $\frac{s}{\sqrt{N}}$: $$m = t^* \times SE$$ This means that you can compute the standard error $SE$ if you know $t^*$ and the margin of error $m$. Can you do it?

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