The left-most green line is erected where z=-1.645. The right-most green line is erected where z=+1.645. The area bounded by the curve, the z-axis, and those two green lines is 90% of the total area between the curve and the z-axis. The area to the left of the left green line is 5% of the total area, and the area to the right of the right green line is the other 5% of the total area between the curve and the z-axis. Your are testing the hypothesis H0: m = 500 against the alternative hypothess Ha: m ≠ 500 Suppose you took a sample of size n=100 and found that the sample mean was ⴳ = 493 and the standard deviation were s = 50 Then you would calculate this test statistic: ⴳ - m z = 覧覧覧覧 493 - 500 z = 覧覧覧覧覧覧 = -1.4 This would correcpond to the red line drawn at -1.4 and since the 1.4 value of z is between the two green bars, the mean of 493 is close enough to 500 with a sample that size with that standard deviation, and so we would fail to reject the hypothesis. However, suppose you took a sample of size n=64 and found that the sample mean was ⴳ = 495 and the standard deviation were s = 20 Then you would calculate this test statistic: ⴳ - m z = 覧覧覧覧 495 - 500 z = 覧覧覧覧覧覧 = -2.0 This would correcpond to the red line drawn at -2.0 and since the 2.0 value of z is NOT between the two green bars, the mean of 495 with a sample that size with that standard deviation, is not close enough to 500, and so we would reject the null hypothesis. Notice that a lot depends on the size and the standard deviation of the sample we take. Now since this is a 2-tail test, the sample mean could be larger than 500 Suppose you took a sample of size n=81 and found that the sample mean was ⴳ = 510 and the standard deviation were s = 70 Then you would calculate this test statistic: ⴳ - m z = 覧覧覧覧 510 - 500 z = 覧覧覧覧覧覧 = -1.29 This would correcpond to the red line drawn at -1.29 and since the 1.29 value of z is between the two green bars, the mean of 510 is close enough to 500 with a sample that size with that standard deviation, and so we would fail to reject the hypothesis. However, suppose you took a sample of size n=150 and found that the sample mean was ⴳ = 506 and the standard deviation were s = 40 Then you would calculate this test statistic: ⴳ - m z = 覧覧覧覧 506 - 500 z = 覧覧覧覧覧覧 = 1.84 This would correcpond to the red line drawn at +1.84 and since the 1.84 value of z is NOT between the two green bars, the mean of 506 with a sample that size with that standard deviation, is not close enough to 500, and so we would reject the null hypothesis. Edwin