Question

In: Statistics and Probability

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1

1) Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1 (0.9616)

2) Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. The shaded area is z=0.98 to the left.

Solution

Step #1

Normal distribution: Normal distribution is a continuous distribution of data that has the bell-shaped curve. The normally \((\mu)\) distributed random variable \(x\) has mean \(\left({ }^{\mu}\right)\) and standard deviation \((\sigma)\). Also, the standard normal distribution represents a normal curve with mean 0 and standard deviation 1.
Thus, the parameters involved in a normal distribution are mean \((\mu)\) and standard deviation \((\sigma)\) Standardized z-score: The standardized z-score represents the number of standard deviations the data point is away from the mean.
- If the z-score takes a positive value when it is above the mean (0).
- If the z-score takes a negative value when it is below the mean (0).
Let \(X \sim N(\mu, \sigma)\), then the standard z-score is found using the formula given below: $$z=\frac{X-\mu}{\sigma}$$ Where \(X\) denotes the individual raw score, \(\mu\) denotes the population means and \(\sigma\) denotes the population standard deviation. The procedure for finding the z-value is listed below:
- From the table of the standard normal distribution, locate the probability value. Move left until the first column is reached.
- Move upward until the top row is reached.
\(\cdot\) Locate the probability value, by the intersection of the row and column values give the area to the left of z. Procedure for finding the probability value (confidence level) from standard normal table: In the standard normal table first locate the value \(0 . \mathrm{a}\) in the first \(\mathrm{z}\) column. \(\cdot\) Locate the value of \(0.0 \mathrm{b}\) in the first \(z\) row.
- Move right until the column of \(0.0 \mathrm{b}\) is reached.
- Move down until the row \(0 . \mathrm{a}\) is reached. Locate the probability value, by the intersection of the row and column values, gives the area to the left of z.

(1)

From the information, it is clear that the graph depicts the standard normal distribution with mean 0 and standard deviation 1 with the probability 0.9616.

The procedure for finding the z-value is listed below:

1.From the table of the standard normal distribution, locate the probability value as 0.9616.

2.Move left until the first column is reached. Note as 1.7.

3.Move upward until the top row is reached. Note as 0.07.

4.Locate the probability value, by the intersection of the row and column values, gives the area to the left of z = 1.77 

(2)

The probability\(P(z<0.98)\)  is obtained below:

Procedure for finding the probability value (confidence level) from the standard normal table:

Let the z value be 0.98.

• In the standard normal table first locate the value 0.9 in the first z column.

• Locate the value of 0.08 in the first z row.

• Move right until the column of 0.08 is reached.

• Move down until row 0.9 is reached.

• Locate the probability value, by the intersection of the row and column values gives the area to the left of z = 0.98

The probability value is 0.8365.

Part 1 Answer

The indicated z-score is \(1.77 .\)

Part 2 Answer

The probability \(P(z<0.98)\) is \(0.8365 .\)