Sketch a standard normal curve and shade the area that lies to the right of [tex]$a . -1.23, b . 0.5, c . 0$[/tex], and [tex]$d . 4 . 2$[/tex]. Then determine the area under the standard normal. The area to the right of 0 is $\square$ $\downarrow$. (Round to four decimal places as needed.)
Answer (2)
To find the area to the right of specified z-scores under the standard normal curve, we calculated the areas for z = -1.23 (0.8907), z = 0.5 (0.3085), z = 0 (0.5000), and z = 4.2 (0.0000). The area to the right of 0 is 0.5000. Each area was found by subtracting the area to the left from 1, following the properties of the standard normal distribution.
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Calculate the area to the right of z = -1.23: 1 − P ( Z < − 1.23 ) = 0.8907 .
Calculate the area to the right of z = 0.5: 1 − P ( Z < 0.5 ) = 0.3085 .
Calculate the area to the right of z = 0: 1 − P ( Z < 0 ) = 0.5000 .
Calculate the area to the right of z = 4.2: 1 − P ( Z < 4.2 ) = 0.0000 .
The area to the right of 0 is 0.5000 .
Explanation
Understand the problem and provided data We are asked to find the area to the right of given values under the standard normal curve. The standard normal curve has a mean of 0 and a standard deviation of 1. We need to find the area to the right of z = -1.23, z = 0.5, z = 0, and z = 4.2. We are asked to round the area to four decimal places.
Area to the right To find the area to the right of a z-score, we subtract the area to the left of the z-score from 1. This is because the total area under the standard normal curve is 1.
Calculate area to the right of -1.23 For z = -1.23, the area to the right is -1.23) = 1 - P(Z < -1.23)"> P ( Z > − 1.23 ) = 1 − P ( Z < − 1.23 ) . Using a standard normal table or calculator, we find that P ( Z < − 1.23 ) ≈ 0.1093 . Therefore, -1.23) = 1 - 0.1093 = 0.8907"> P ( Z > − 1.23 ) = 1 − 0.1093 = 0.8907 .
Calculate area to the right of 0.5 For z = 0.5, the area to the right is 0.5) = 1 - P(Z < 0.5)"> P ( Z > 0.5 ) = 1 − P ( Z < 0.5 ) . Using a standard normal table or calculator, we find that P ( Z < 0.5 ) ≈ 0.6915 . Therefore, 0.5) = 1 - 0.6915 = 0.3085"> P ( Z > 0.5 ) = 1 − 0.6915 = 0.3085 .
Calculate area to the right of 0 For z = 0, the area to the right is 0) = 1 - P(Z < 0)"> P ( Z > 0 ) = 1 − P ( Z < 0 ) . Since the standard normal curve is symmetric around 0, P ( Z < 0 ) = 0.5 . Therefore, 0) = 1 - 0.5 = 0.5"> P ( Z > 0 ) = 1 − 0.5 = 0.5 .
Calculate area to the right of 4.2 For z = 4.2, the area to the right is 4.2) = 1 - P(Z < 4.2)"> P ( Z > 4.2 ) = 1 − P ( Z < 4.2 ) . Using a standard normal table or calculator, we find that P ( Z < 4.2 ) ≈ 1.0000 . Therefore, 4.2) = 1 - 1.0000 = 0.0000"> P ( Z > 4.2 ) = 1 − 1.0000 = 0.0000 .
State the area to the right of 0 The area to the right of 0 is 0.5000.
Examples
Understanding areas under the normal curve is crucial in many real-world scenarios. For instance, in quality control, manufacturers use these areas to determine the probability that a product's measurement falls within acceptable limits. Similarly, in finance, these probabilities help assess the risk associated with investments by estimating the likelihood of returns falling within certain ranges. These calculations provide a foundation for making informed decisions based on statistical probabilities.
