Answer :
First we need to find the mean and the standadr deviation. For the mean, we sum the values up and divide by the number of data given. We have 8 data, so:
[tex]\mu=\frac{93+65+67+67+67+65+66+65}{8}=\frac{555}{8}=69.375[/tex]Now, we need the standard deviation. To find it, we use the formula:
[tex]\sigma=\sqrt[]{\frac{\sum ^{}_{}(x_i-\mu)^2}{N-1}}[/tex]The sum symbol in there mean we need to get each data given, substract the mean we calculated, square them and then sum all. N is the number of data, which is 8. Let look one example:
For data 93, we do:
[tex](x_i-\mu)^2=(93-69.375)^2=23.625^2=558.141[/tex]Now, we do the same for the others:
[tex]\begin{gathered} (x_i-\mu)^2=(65-69.375)^2=(-4.375)^2=19.141 \\ (x_i-\mu)^2=(67-69.375)^2=(-2.375)^2=5.641 \\ (x_i-\mu)^2=(67-69.375)^2=(-2.375)^2=5.641 \\ (x_i-\mu)^2=(67-69.375)^2=(-2.375)^2=5.641 \\ (x_i-\mu)^2=(65-69.375)^2=(-4.375)^2=19.141 \\ (x_i-\mu)^2=(66-69.375)^2=(-3.375)^2=11.391 \\ (x_i-\mu)^2=(65-69.375)^2=(-4.375)^2=19.141 \end{gathered}[/tex]Now we sum all of them:
[tex]558.141+19.141+5.641+5.641+5.641+19.141+11.391+19.141=643.878[/tex]This goes into the formula.
[tex]\sigma=\sqrt[]{\frac{643.878}{8-1}}=\sqrt[]{\frac{643.878}{7}}=\sqrt[]{91.982}=9.591[/tex]Now, we want the number of data that is within 2 standard deviation from the mean. Thus, we want the data that are between:
[tex]\begin{gathered} \mu-2\sigma=69.375-2\cdot9.591=50.193 \\ \mu+2\sigma=69.375+2\cdot9.591=88.557 \end{gathered}[/tex]From the given Data, 93 is out of this range, but all the others (65, 67, 67, 67, 65, 66, 65) are in it. So, the number of data within 2 standard deviations from the mean is 7.