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Chapter 6: Problem 55

Determine the value of \(z\) so that the area under the standard normal curve a. in the right tail is \(.0500\) b. in the left tail is \(.0250\) c. in the left tail is \(.0100\) d. in the right tail is \(.0050\)

Short Answer

Expert verified
The corresponding \(z\) values for the areas under the standard normal curve are: a. \(1.64\), b. \(-1.96\), c. \(-2.33\), and d. \(2.57\).

Step by step solution

01

Understand the standard normal distribution

The standard normal distribution is a special case of the normal distribution where the mean is zero and the standard deviation is 1. The skewness and kurtosis are also standardized, resulting in a bell-shaped graph or curve. The total area under this curve is 1.
02

Interpret the request for right tail or left tail

The 'right tail' refers to the part of the distribution curve that lies to the right of the \(z\) value, while the 'left tail' refers to the part of the curve that lies to the left of the \(z\) value. The right tail includes all values greater than \(z\), while the left tail includes all values less than \(z\). In a standard normal distribution curve, the area in the right tail corresponds to the probability that a randomly chosen standard normal variable is greater than \(z\). Similarly, the area in the left tail corresponds to the probability that a random standard normal variable is less than \(z\).
03

Lookup the z-table in reverse or use the inverse Z-table function.

Normally, we use the table of standard normal distribution or z-table to find the probability given a \(z\) value. However, in this case, we have the area under the curve (probability) specified, and we need to find the corresponding \(z\) value(s). Thus, we need to do a reverse lookup of the z-table or use the inverse function of Z-table lookup on software programs such as Excel, MATLAB or R.
04

Find the corresponding \(z\) value for each area.

a. For right tail area \(.0500\), the \(z\) value is approximately \(1.64\) (as the cumulative probability on left is \(.9500\)).b. For left tail area \(.0250\), the \(z\) value is approximately \(-1.96\) (as the cumulative probability on left is \(.0250\)).c. For left tail area \(.0100\), the \(z\) value is approximately \(-2.33\) (as the cumulative probability on left is \(.0100\)).d. For right tail area \(.0050\), the \(z\) value is approximately \(2.57\) (as the cumulative probability on left is \(.9950\)).

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Most popular questions from this chapter

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