Consider the following data set:                      1, 1, 3, 4, 5, 6, 7, 7, 8, 11, 12, 13, 13, 14, 15, 15How many of the values are within one standard deviation of the mean?

To answer this question, we first need to calculate the mean (average) and the standard deviation of the data set.

Step 1: Calculate the mean The mean is the sum of all the values divided by the number of values.

1+1+3+4+5+6+7+7+8+11+12+13+13+14+15+15 = 135

There are 16 values in the data set.

So, the mean is 135/16 = 8.4375

Step 2: Calculate the standard deviation First, subtract the mean from each value to get the deviation of each value. Then square each deviation. Add up all these squared values and divide by the number of values to get the variance. The standard deviation is the square root of the variance.

The deviations are: -7.4375, -7.4375, -5.4375, -4.4375, -3.4375, -2.4375, -1.4375, -1.4375, -0.4375, 2.5625, 3.5625, 4.5625, 4.5625, 5.5625, 6.5625, 6.5625

The squared deviations are: 55.3164, 55.3164, 29.5664, 19.6914, 11.8164, 5.9414, 2.0664, 2.0664, 0.1914, 6.5664, 12.6914, 20.8164, 20.8164, 30.9414, 43.0664, 43.0664

The sum of the squared deviations is: 359.25

The variance is 359.25/16 = 22.453125

The standard deviation is the square root of the variance, which is approximately 4.74.

Step 3: Determine how many values are within one standard deviation of the mean This means we are looking for values that are between (mean - standard deviation) and (mean + standard deviation), or between 8.4375 - 4.74 and 8.4375 + 4.74, which is between 3.6975 and 13.1775.

Looking at the data set, the values within this range are: 4, 5, 6, 7, 7, 8, 11, 12, 13, 13

So, 10 of the values are within one standard deviation of the mean.