**Find the corresponding score for the given observation and select the right one from the given options:**

**a) 0.97**

**b) -0.97**

**c) 0.64**

**d) -0.97**

The objective of this question is to find the **corresponding score** of **normal distribution** for the given observation.

This question uses the concept of the **Normal distribution** to find the **corresponding score** for the given **observation**. The normal distribution is **symmetric** near the **mean** which shows that the point from the data near the mean occurs more frequently. The normal distribution has the **shape** of the **bell curve** in the graph.

## Expert Answer

Given that the** observation** $x$ is $470$.

**mean**, $\mu$ is $570$.

and the **standard deviation**, $\sigma$ is $103$.

For the occurrence score $z$, we have the **formula** given below as:

\[z=\frac{x-\mu}{\sigma}\]

where $x$ is the given **observation**, \mu is the** mean**, and \sigma is the** standard deviation**.

By putting the **values** of observation, the mean, and standard deviation in the above formula, we get:

\[z=\frac{470-570}{103}\]

In the above step, we **subtracted** the value of observation from the occurrence, and this results in:

\[z=\frac{-100}{103}\]

\[z=-0.97\]

So the **correct** answer is $-0.97$.

## Numerical Result

The **occurrence score** for the observation $x=470$, $\mu 570$ and $\sigma 103$ is $-0.97$.

## Example

Find the occurrence score for the observation of $10$,$50$,$100$, and $200$ when the mean, $\mu$ is 400 and the standard deviation, \sigma is 200.

From the **given data**, we know that:

**observation** $x$ is $10$, $100$, $200$ and $50$.

**mean**,$\mu$ is $400$ .

and **standard deviation**,$\sigma$ is $200$. To find the **occurrence score** we have the formula given below as:

\[z=\frac{x-\mu}{\sigma}\]

$x$ is the given observation, \mu is the mean, and \sigma is the standard deviation.

First, we will calculate the **occurrence score** for the observation value of $10$.

\[z=\frac{10-400}{200}\]

\[z=\frac{-390}{200}\]

By **simplifying** it, we get:

\[z=-1.95\]

Hence the **occurrence score** for observation $10$, $\mu 400$ and $\sigma 200$ is $-1.95$

Now to calculate the occurrence score for **observation** $50$, we have the formula:

\[z=\frac{x-\mu}{\sigma}\]

By putting values in the above **formula**, we get:

\[z=\frac{50-400}{200}\]

\[z=\frac{-350}{200}\]

Thus, **simplifying** it results in:

\[z=-1.75\]

Now calculate the occurrence score for **observation** $100$. The **formula** is:

\[z=\frac{x-\mu}{\sigma}\]

\[z=\frac{100-400}{200}\]

\[z=\frac{-300}{200}\]

Hence, simplifying it **results** in:

\[z=-1.5\]

and for **observation** of $200$, we use the formula :

\[z=\frac{x-\mu}{\sigma}\]

\[z=\frac{200-400}{200}\]

\[z=\frac{-200}{200}\]

Therefore, simplifying it **results** in:

\[z=-1\]

Therefore, we have calculated the **o****ccurrence score** for the **different** values of **observation** while the values of **mean** and **standard deviation** remain the **same**.