Question

Determine the following standard normal (z) curve areas: a. The area under the \(z\) curve to the left of \(1.75\) b. The area under the \(z\) curve to the left of \(-0.68\) c. The area under the \(z\) curve to the right of \(1.20\) d. The area under the \(z\) curve to the right of \(-2.82\) e. The area under the \(z\) curve between \(-2.22\) and \(0.53\) f. The area under the \(z\) curve between \(-1\) and 1 \(\mathrm{g}\). The area under the \(z\) curve between \(-4\) and 4

Short answer

a) The area under the z curve to the left of 1.75 is approximately 0.9599.\n b) The area under the z curve to the left of -0.68 is approximately 0.2483.\n c) The area under the z curve to the right of 1.20 is approximately 0.1151.\n d) The area under the z curve to the right of -2.82 is approximately 0.9974.\n e) The area under the z curve between -2.22 and 0.53 is approximately 0.8654.\n f) The area under the z curve between -1 and 1 is approximately 0.6826.\n g) The area under the z curve between -4 and 4 is approximately 1.

Step-by-step solution

Calculate the area to the left of a given z-value

To calculate the area under the curve to the left of a given z-value, simply look up the z-value in a standard normal (z) table. This directly gives the proportion of values to the left of the given z-value. For example, the area to the left of \(1.75\) is approximately \(0.9599\) or \(95.99\%\) of the total area under the curve.

Calculate the area to the right of a certain z-value

To calculate the area to the right of a certain z-score, subtract the table value associated with that z-score from 1. This is because the total area under the curve is 1, and the table gives the area to the left of the z-score. For example, the area to the right of \(1.20\) is \(1 - 0.8849 = 0.1151\) or \(11.51\%\).

Calculate the area between two given z-values

To calculate this area, find the area to the left of the higher z-value, and subtract the area to the left of the lower z-value from it. For instance, the area between \(-1\) and \(1\) is \(0.8413 - 0.1587 = 0.6826\) or \(68.26\%\).