Question

A variable is normally distributed with mean 20 and standard deviation 5 . Use your graphing calculator to find each of the following areas. Write your answers in decimal form. Round to the nearest thousandth as needed. a) Find the area to the left of 23. b) Find the area to the left of 15. c) Find the area to the right of 18. d) Find the area to the right of 26 . e) Find the area between 15 and 29.

Answer 1

To find the areas, we use the z-score formula and the standard normal distribution. For each question, we calculate the z-score and find the corresponding area using a calculator or the standard normal distribution table. The areas to the left of 23, 15, and 18 are approximately 0.7257, 0.1587, and 0.3446, respectively. The area to the right of 26 is approximately 0.1151, and the area between 15 and 29 is approximately 0.7262.

Step-by-step explanation:

To find the areas in question, we can use the standard normal distribution and the z-score formula. The z-score formula is:

z = (x - mean) / standard deviation

Let's calculate the areas for each question:

a) Find the area to the left of 23:

z = (23 - 20) / 5 = 0.6

Using the calculator or the standard normal distribution table, we can find that the area to the left of 0.6 is approximately 0.7257.

b) Find the area to the left of 15:

z = (15 - 20) / 5 = -1

The area to the left of -1 is approximately 0.1587.

c) Find the area to the right of 18:

z = (18 - 20) / 5 = -0.4

The area to the right of -0.4 is 1 - 0.6554 = 0.3446.

d) Find the area to the right of 26:

z = (26 - 20) / 5 = 1.2

The area to the right of 1.2 is 1 - 0.8849 = 0.1151.

e) Find the area between 15 and 29:

The area to the left of 15 is 0.1587 and the area to the left of 29 is 0.8849. Therefore, the area between 15 and 29 is 0.8849 - 0.1587 = 0.7262.

Answer 2

To find the areas using a graphing calculator, convert the given values to z-scores and use a graphing calculator or calculator to find the required areas.

Step-by-step explanation:

To find the areas using a graphing calculator, we need to convert the given values to z-scores. The z-score formula is z = (x - mean) / standard deviation. Using this formula, we can calculate the z-scores for each value.

a) To find the area to the left of 23, we need to find the z-score for 23. z = (23 - 20) / 5 = 0.6. Using a graphing calculator, we can find that the area to the left of a z-score of 0.6 is approximately 0.725.

b) To find the area to the left of 15, we need to find the z-score for 15. z = (15 - 20) / 5 = -1. Using a graphing calculator, we can find that the area to the left of a z-score of -1 is approximately 0.158.

c) To find the area to the right of 18, we can find the area to the left of 18 and subtract it from 1. To find the area to the left of 18, we need to find the z-score for 18. z = (18 - 20) / 5 = -0.4. Using a graphing calculator, we can find that the area to the left of a z-score of -0.4 is approximately 0.344. Therefore, the area to the right of 18 is 1 - 0.344 = 0.656.

d) To find the area to the right of 26, we need to find the z-score for 26. z = (26 - 20) / 5 = 1.2. Using a graphing calculator, we can find that the area to the right of a z-score of 1.2 is approximately 0.115.

e) To find the area between 15 and 29, we need to find the areas to the left of 15 and to the left of 29 and subtract the latter from the former. Using the z-scores calculated earlier, the area to the left of 15 is approximately 0.158 and the area to the left of 29 is approximately 0.884.

Therefore, the area between 15 and 29 is 0.884 - 0.158 = 0.726.