Question

the scores of 8 grade students in math test are normal distributed with a mean of 57.5 abs a standard deviation of 6.5 from thus data, we can conclude that 68% of the students received scores between 34 or 51 and 64 or 68.

Answer 1

We need to calculate the Z-score of the given values.

I'll draw the normal distribution curve for this example

Z-score of 34

`\begin{gathered} Z=(x-\mu)/(\sigma) \\ Z=(34-57.5)/(6.5)=-3.615 \end{gathered}`

If we use a Z table, we can see that a value of Z=-3.615 corresponds to an area of 0.00015

Z-score of 51

`Z=(51-57.5)/(6.5)=-1`

Again, using the Z table, when Z=-1, the area below the curve is 0.15866.

If we substract 0.15866 and 0.00015, we obtain the area between 51 and 34, as shown below

`0.15866-0.00015=0.15851`

In percentage: 15.851%

Now, let's proceed with 64 and 68

Z-score of 64:

`Z=(64-57.5)/(6.5)=1`

This value represents an area to the left of: 0.84134

Z-score of 68:

`Z=(68-57.5)/(6.5)=1.62`

which corresponds to an area to the left of: 0.94738

Again, we must substract those two values to obtain the areal between 68 and 64

`0.94738-0.84134=0.10604`

In percentage, this is equal to 10.604%

I will upload an image of Z table in case you want to double check...

Now, in order to conclude the question... we need to check if "68% of the students received scores between 34 or 51 and 64 or 68."

We already know that between 34 and 51 there are 15.851% of students and between 64 and 68 there are 10.604%, which adds a value of 26.455%

In conclusion, the information given is wrong, because it is 26.455% and not 68%