Range Size: 1 RANGE ROOT COST a..a a .09 and..and he..he and he .13 .07 I..I I .14 it..it it .07 not..not not .10 not..or not .24 or..or or .07 she..she the..the you..you she the you .07 .15 .11 Range Size: 2 RANGE ROOT COST a..and and .31 and..he and .27 he..I I .28 I..it I .28 it..not not .24 or..she she..the the..you or the the .21 .29 .37 and..I he .61 he..it I .42 I..not it .55 it..or not .38 and..it I .75 he..not I .69 I..or not .73 it..she not .59 not..the or..you or the .78 .72 and..not I 1.02 he..or I .90 I..she not .94 it..the or .99 not..you the 1.02 Range Size: 3 RANGE ROOT COST a..he and .45 not..she or..the she..you or the the .41 .50 .51 Range Size: 4 RANGE ROOT COST a..I and .80 Range Size: 5 RANGE ROOT COST a..it and 1.01 Range Size: 6 RANGE ROOT COST a..not I 1.29 Range Size: 7 and..or not 1.40 he..she not 1.15 Range Size: 8 RANGE ROOT COST CS 340 a..she I 1.78 I..the not 1.38 it..you the 1.27 RANGE ROOT COST a..or I 1.50 Range Size: 9 and..the he..you not not 1.92 2.05 RANGE ROOT COST a..the and..you I not 2.38 2.33 and..she he..the I not 1.59 1.51 I..you not 1.71 Range Size: 10 RANGE ROOT COST a..you I 2.72 Page 12

The sampling distribution of When we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts. MEAN AND STANDARD DEVIATION OF A SAMPLE MEAN Suppose that is the mean of an SRS of size drawn from a large population with mean and standard deviation . Then the sampling distribution of has mean and standard deviation . We say the statistic is an unbiased estimator of the parameter . Because it’s standard deviation is , the averages are less variable than individual observations, and the results of large samples are less variable than the results of small samples. SAMPLING DISTRIBUTION OF A SAMPLE MEAN If individual observations have the distribution, then the sample mean of an SRS of size has the distribution.

Standard Deviation = SD • Standard Deviation uses all the data points, not just some like Range and Interquartile Range • Standard Deviation does not have squared units (like Variance) and is thus easier to interpret • Standard deviation has the same units as the data!! • The sample standard deviation is a point estimator of the population standard deviation • Interpretation of Standard Deviation: • • • • • • • A Numerical Measure that says how much variability there is in the data points Standard Deviation Is Like An Average Of The Deviations Standard Deviation tells us how fairly the mean represents its data points Standard Deviation tells us how clustered the data points are around the mean For financial assets standard deviation is a measure of risk or fluctuation in asset value Use STDEV.P function for population data Use STDEV.S for sample data. 56

Does it look like a Normal distribution? • The variance of a measured speed distribution normally should be less than the variance of a random distribution (estimated by the sample mean) • The standard deviation should be approximately one-sixth of the total range since the mean plus and minus three standard deviations encompasses 99.73 percent of the observations of a normal distribution • The standard deviation should be approximately one-half of the 15 to 85 percentile range since the mean plus and minus one standard deviation encompasses 68.27 percent of the observation of a normal distribution, • The 10-mph pace should be spread equally on each side of the mean • The normal distribution has little skewness and the coefficient of skewness should be approximately zero. mean - mode mean - median coef skew = = 3( ) s s Example on pages 102,103

Variability indices 1) Range-total number of possible values between the minimum and maximum values. 2) Variance- the average squared deviation of each number from the mean 3) Standard deviation- the square root of the variance. Tells us how far the scores in a distribution are deviating from the mean, on average. • An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score. • In a normal distribution, about 68% of the scores are within one standard deviation of the mean and about 95% of the scores are within two standard deviations of the mean

Exercise solution // Removes the given value from this BST, if it exists. public void remove(int value) { overallRoot = remove(overallRoot, value); } private IntTreeNode remove(IntTreeNode root, int value) { if (root == null) { return null; } else if (root.data > value) { root.left = remove(root.left, value); } else if (root.data < value) { root.right = remove(root.right, value); } else { // root.data == value; remove this node if (root.right == null) { return root.left; // no R child; replace w/ L } else if (root.left == null) { return root.right; // no L child; replace w/ R } else { // both children; replace w/ min from R root.data = getMin(root.right); root.right = remove(root.right, root.data); } } return root; } 41

Exercise solution // Removes the given value from this BST, if it exists. public void remove(int value) { overallRoot = remove(overallRoot, value); } private IntTreeNode remove(IntTreeNode root, int value) { if (root == null) { return null; } else if (root.data > value) { root.left = remove(root.left, value); } else if (root.data < value) { root.right = remove(root.right, value); } else { // root.data == value; remove this node if (root.right == null) { return root.left; // no R child; replace w/ L } else if (root.left == null) { return root.right; // no L child; replace w/ R } else { // both children; replace w/ min from R root.data = getMin(root.right); root.right = remove(root.right, root.data); } } return root; } 52

The Sampling Distribution of x 12 When we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts. The Sampling Distribution of Sample Means Suppose that x is the mean of an SRS of size n drawn from a large population with mean and standard deviation s . Then : The mean of the sampling distribution of x is x = The standard deviation of the sampling distribution of x is s sx = n Note: These facts about the mean and standard deviation of x are true no matter what shape the population distribution has. If individual observations have the N(µ,σ) distribution, then the sample mean of an SRS of size n has the N(µ, σ/√n) distribution regardless of the sample size n.