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X~N(570, 103). Find the z-score corresponding to an observation of 470.

  • 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 occurrence score for the different values of observation while the values of mean and standard deviation remain the same.

5/5 - (17 votes)