Sketch a standard normal curve and shade the area that lies to the right of a. -1.23, b. 0.5, c. 0, and d. 4.2. Then determine the area under the standard normal curve that lies in the area of interest for each part.
The area to the right of -1.23 is
(Round to four decimal places as needed.)
Answer (1)
The area to the right of a z-score is calculated by subtracting the area to the left of the z-score from 1.
For z = -1.23, the area to the right is 1 − 0.1093 = 0.8907 .
For z = 0.5, the area to the right is 1 − 0.6915 = 0.3085 .
For z = 0, the area to the right is 1 − 0.5 = 0.5 , and for z = 4.2, the area to the right is approximately 1 − 1.0000 = 0.0000 .
The area to the right of -1.23 is 0.8907 .
Explanation
Understand the problem and provided data We are asked to find the area under the standard normal curve to the right of given z-values. The standard normal curve is a bell-shaped curve with a mean of 0 and a standard deviation of 1. The area under the curve represents probability, and the total area under the curve is equal to 1.
Area to the right calculation To find the area to the right of a given z-score, we first find the area to the left of that z-score using a standard normal table or a calculator with statistical functions. Then, we subtract the area to the left from 1 to find the area to the right, since the total area under the curve is 1. The formula is: Area to the right = 1 - Area to the left.
Calculate the area to the right of z = -1.23 a. For z = -1.23, the area to the left is approximately 0.1093. Therefore, the area to the right is 1 − 0.1093 = 0.8907 .
Calculate the area to the right of z = 0.5 b. For z = 0.5, the area to the left is approximately 0.6915. Therefore, the area to the right is 1 − 0.6915 = 0.3085 .
Calculate the area to the right of z = 0 c. For z = 0, the area to the left is 0.5. Therefore, the area to the right is 1 − 0.5 = 0.5 .
Calculate the area to the right of z = 4.2 d. For z = 4.2, the area to the left is approximately 1.0000. Therefore, the area to the right is 1 − 1.0000 = 0.0000 .
State the final answer The areas to the right of the given z-scores are: a. z = -1.23: 0.8907 b. z = 0.5: 0.3085 c. z = 0: 0.5000 d. z = 4.2: 0.0000
Examples
Understanding areas under the normal curve is crucial in many real-world scenarios. For instance, in quality control, it helps determine the probability that a manufactured product meets certain specifications. In finance, it's used to assess the risk associated with investments. By calculating these probabilities, businesses can make informed decisions and minimize potential losses. For example, a company might use the area to the right of a certain z-score to estimate the likelihood that a product's weight falls within an acceptable range, ensuring customer satisfaction and regulatory compliance.
