Question

assume z is a standard normal random variable. what is the value of z if the area between -z and z is 0..754

Answer 1

To find the value of z, use the invNorm(0.877,0,1) command or the standard normal distribution table to find a z-score of approximately 1.17.

Step-by-step explanation:

To find the value of z if the area between -z and z is 0.754, we can use the standard normal distribution table or a calculator command like invNorm(0.877,0,1). This command will yield a z-score of approximately 1.17, which means that for a standard normal distribution, the area between -1.17 and 1.17 is 0.754.

Answer 2

To find the z-value when the area between -z and z is 0.754, calculate the area to the left of z using 1 - (1 - 0.754) / 2 and use the invNorm function or a z-table to find the corresponding z-score.

Step-by-step explanation:

The area between -z and z represents the central portion of the standard normal distribution. Since we know the area between -z and z is 0.754, the total area in the tails will be 1 - 0.754 = 0.246, which is divided equally across both tails, giving us an area of 0.123 in each tail.

Using the inverse normal function on your calculator (invNorm), you should input the area to the left of z, which is 1 - 0.123 = 0.877. The TI-83, 83+, or 84+ calculator command is invNorm(0.877, 0, 1) as the mean (0) and standard deviation (1) for the standard normal distribution are used by default. The output will give you the positive z-value that leaves an area of 0.877 to the left, which, due to the symmetry of the standard normal distribution, will be the same as the absolute value of the negative z.

To find z using a standard normal probability table, look for the value closest to 0.877 in the table. The corresponding z-score will be your value. If the table does not have the exact values, you may need to interpolate between the closest values to get a more accurate answer.