Let Z Follow The Standard Normal Distribution Show Your Manu

Let Z Follow The Standard Normal Distributionshow Your Manual Wo

(1) Let Z follow the standard normal distribution. Show your manual work to find z such that P(Z > z) = 0.95. (Hint: we need to find the left area first then inversely find the z; be careful about the sign of z-value.)

(2) Let Z follow the standard normal distribution. Show your Excel stats function to find z such that P(Z > z) = 0.95. (The same hint above.)

(3) Let Z follow the standard normal distribution. Show your manual work to find z such that P(-z

(4) Let Z follow the standard normal distribution. Show your Excel stats function to find z such that P(-z

(5) Let X follow a normal distribution with a mean of 10 and a standard deviation of 2. Show your manual work to find x such that P(X

(6) Let X follow a normal distribution with a mean of 10 and a standard deviation of 2. Show your Excel stats functional work to find x such that P(X

(7) Let X follow a normal distribution with a mean of 10 and a standard deviation of 2. Show your manual work to find x such that P(X > x) = 0.05.

(8) Let X follow a normal distribution with a mean of 10 and a standard deviation of 2. Show your Excel stats functional work to find x such that P(X > x) = 0.05.

Paper For Above instruction

The following analysis explores the process of determining specific z-scores and x-values associated with given probabilities in the context of the standard normal distribution and a normal distribution with a specified mean and standard deviation. Both manual calculations and utilization of Excel statistical functions are demonstrated to understand the procedures and to enable practical application in statistical analysis.

Manual Calculation of z-Score for P(Z > z) = 0.95

To find the z-value such that P(Z > z) = 0.95 in a standard normal distribution, we need to determine the corresponding lower-tail probability because the standard normal table typically provides the cumulative probability from the far left up to the z-score. Since P(Z > z) = 0.95, the probability that Z is greater than z is 0.95, meaning that the probability that Z is less than z is 1 - 0.95 = 0.05.

Thus, we seek z such that P(Z

Hence, the manual value of z satisfying P(Z > z) = 0.95 is approximately -1.645, keeping in mind the sign conventions.

Excel Function to Find z for P(Z > z) = 0.95

Using Excel, the function NORM.S.INV (or NORMSINV in older versions) returns the inverse of the standard normal cumulative distribution function. To find z such that P(Z > z) = 0.95, we first find the corresponding lower-tail probability: 1 - 0.95 = 0.05. Therefore, we input:

=NORM.S.INV(0.05)

which yields approximately -1.645, consistent with manual calculations.

Finding z for P(-z

The probability that Z falls between -z and z is 0.95, implying that 5% is split equally in the two tails: 2.5% in each tail. Therefore, the cumulative probability up to z (on the positive side) is 1 - 0.025 = 0.975.

From standard normal tables or Excel, the z-score corresponding to a cumulative probability of 0.975 is approximately 1.96. Thus, z ≈ 1.96.

Manual Calculation of z for P(-z

As outlined above, z ≈ 1.96 gives the symmetric bounds for capturing 95% of the distribution centered at zero.

Finding x such that P(X

Given X ~ N(μ=10, σ=2), and P(X

Using the z-score formula:

x = μ + z  σ = 10 + 1.645  2 = 10 + 3.29 = 13.29

Therefore, x ≈ 13.29.

Excel Function to Find x such that P(X

Using Excel's NORM.INV function:

=NORM.INV(0.95, 10, 2)

This function directly computes x, yielding approximately 13.29.

Finding x such that P(X > x) = 0.05 (Manual Calculation)

Since P(X > x) = 0.05, P(X

Excel Function to Find x such that P(X > x) = 0.05

Similarly, use:

=NORM.INV(0.95, 10, 2)

to obtain the same x-value, approximately 13.29.

Conclusion

The combination of manual calculations using z-scores and the use of Excel functions provides accurate tools for statistical analysis. Understanding the relation between probabilities and z-values/quantiles in the normal distribution is essential for many applications in research, data analysis, and decision-making.

References

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