Find The Value Of Z For The Shaded Region

Find the value of z for which the area of the shaded region under the standard normal curve is as specified

Calculate the z-score corresponding to specific areas (probabilities) under the standard normal distribution curve. For each problem, identify the z-value that yields the given area (area to the left of z) or the area of a shaded region under the standard normal curve.

Sample Paper For Above instruction

The standard normal distribution is symmetric about zero, with a mean of 0 and a standard deviation of 1. It is frequently used in statistics to determine probabilities associated with normal data. Finding the z-score associated with a particular area under the curve involves consulting standard normal distribution tables or using statistical software.

In the first problem, we are asked to find the z-value where the area under the curve is 0.0026. Since the area is small and to the left of z, this corresponds to a very negative z-value. Consulting the z-table or using a calculator, the area of 0.0026 corresponds approximately to z = -2.76. Because the problem might specify whether the area is to the left or right, it's essential to clarify that the standard normal table gives cumulative areas to the left. If the area is to the right, the z-score calculation must adjust accordingly.

Similarly, for an area of 0.0013, which is even smaller and indicates an extreme tail, the z-score is approximately -3.00.

For a shaded area of 0.383, the z-score that leaves this much area to the left is approximately -0.15, since the area to the left of z = -0.15 is about 0.383. Alternatively, if the area is to the right, the z-value would be positive and the calculations adjusted accordingly.

When the area is 0.6915, the corresponding z-score is approximately 0.50, because this z-value cuts off about 69.15% of the distribution to the left.

Understanding how to translate areas into z-scores involves utilizing the inverse cumulative distribution function (inverse norm). This function allows the calculation of z-values given the area to the left under the standard normal curve.

These calculations are fundamental in hypothesis testing, confidence interval construction, and various statistical procedures where probabilities associated with normal distributions are involved. For practice, students should familiarize themselves with standard normal tables and software tools such as statistical calculators or packages like R, Python, or SPSS.

References

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