Final answer:
To find probabilities based on the standard normal variable Z, you calculate the z-score and look up the corresponding area in a z-table. The z-score represents the number of standard deviations a value is from the mean. The area under the normal curve to the left of a z-score corresponds to the probability of getting a value less than that z-score.
Step-by-step explanation:
The probability is represented by the area under the normal curve. To find the probability, calculate the z-score and look up the z-score in the z-table under the z-column. Most z-tables show the area under the normal curve to the left of z. Others show the mean to z area. The method used will be indicated on the table.
You need to find 20.01, having the property that the area under the normal density curve to the right of z0.01 is 0.01 and the area to the left is 0.99. Use your calculator, a computer, or a probability table for the standard normal distribution to find 20.01 = 2.326.
Use the TI-83, 83+, or 84+ calculator command invNorm(0.975,0,1) to find Z0.025. Remember that the area to the right of Z0.025is 0.025, and the area to the left of Z0.025is 0.975. This can also be found using appropriate commands on other calculators, using a computer, or using a standard normal probability table.
We know the mean, standard deviation, and area under the normal curve. We need to find the z-score that corresponds to the area of 0.9 and then substitute it with the mean and standard deviation, into our z-score formula. The z-table shows a z-score of approximately 1.28, for an area under the normal curve to the left of z (larger portion) of approximately 0.9.
Use the Z-table to locate the area under the normal curve to the left of each of these z-scores. The area to the left of the z-score of -0.40 is 0.3446. The area to the left of the z-score of 1.5 is 0.9332. The area between these scores will be the difference in the two areas, or 0.9332 – 0.3446, which equals 0.5886.
Use the TI-83, 83+, or 84+ calculator command invNorm(0.95,0,1) to find Z0.05. Remember that the area to the right of z0.05 is 0.05, and the area to the left of z0.05 is 0.95. This can also be found using appropriate commands on other calculators, using a computer, or using a standard normal probability table. The value of Z0.05 is 1.645.