Exercise 6 Page 743

We want to describe the effect on a normal distribution if each data value increases by 10. To do this, we need to consider how the mean and the standard deviation of the data set will change. Let's start with the mean.

Mean

Assuming that we have n values in our data set, the mean x will be the sum of all of these values divided by n. x=x_1+x_2+...+x_n/n Now, increasing each data value by 10 means that we should add 10 to each term in the numerator. Let's write a new formula for the mean x' x'=(x_1+ 10)+(x_2+ 10)+...+(x_n+ 10)/nNext let's simplify the above expression to see how is it related to the original mean.

Since we have n terms and each of them is increased by 10, we will have the sum of the original values plus n times 10 in the numerator. We can simplify this even further.

Therefore, the new mean is the original mean increased by 10.

Standard Deviation

Now let's recall the formula for the standard deviation of a data set that has n values and mean x. σ=sqrt(∑(x-x)^2/n) We found that when we increase all of the values by 10, the mean will also increase by 10. Using this information, let's write a formula for the new standard deviation σ '. σ '=sqrt(∑((x+ 10)-(x+ 10))^2/n) Let's simplify the expression in the numerator.

We can see that adding a constant to all of the values does not change the standard deviation.

Comparison

Finally, we are ready to describe the effect on the distribution. Since the mean increases by 10, the curve will be translated 10 units to the right.

Notice that the shape remains the same, as the standard deviation did not change.