Quiz 11.2

For a sample of n = 30 scores, X = 45 corresponds to z = 1.50 and X = 40 corresponds to z = +1.00. What are the values for the sample mean and standard deviation?​

First, you want to calculate the distance between the two scores and the corresponding Z-score distance. Therefore: 45-40 = 5 1.5-1 = 0.5 In other words, a z-score of 0.5 is equal to 5 points. We know that one standard deviation equals one z-score. Therefore, one standard deviation (z-score of 1) here would be 2 * 5 = 10 points. To find the mean, simply take one of the X-value, 40 (which is +1 z-score), and subtract one z-score equivalent points from it (10 points). Therefore 40-10 = 30, and 30 is the mean score.

In N = 25 games last season, the college basketball team averaged µ = 76 points with a standard deviation of σ = 6. In their final game of the season, the team scored 89 points. Based on this information, the number of points scored in the final game was ____.​

If a team's average score is 76, and the team got a score of 89, then it's easy to see that the team is performing better than average. But how much better than average is the team performing? This is where your standard deviation comes in. In this problem, the standard deviation is 6. If you subtract 76 from 89, you get a difference of 13 points. You can then divide 13 by 6 to get a z-score of approximately 2.2. How good is a Z-score of +2.2? Please refer to Figure 6.4 on page 166. A Z-score of +2.2 only happens in about 2.28% of the time. What does that look like for a regular NCAA Division 1 team that plays 29 games per season? It means that the chance of a NCAA Division 1 Team having an ultra-performing night with Z-score of +2.2 is only 0.63, or less than one game. In other words, most teams in NCAA Division 1 do not experience this type of superb performance, not even for exhibition games. So yes, a score of 89 for a team with Mean=76, SD=6 statistics is super, ultra, out-of-the world lucky good.

Under what circumstances is a score that is located 5 points above the mean a central value, relatively close to the mean?​

In order to have a score being closer to the mean, you would want a standard deviation that is as large as possible.

Under what circumstances is a score that is 15 points above the mean an extreme score relatively far from the mean?​

To get a score as far away from the mean as possible, you would want to have a standard deviation that is as small as possible.

In a sample with M = 40 and s = 8, what is the z-score corresponding to X = 38?​

Z = (x-M)/s = (38-40)/8 = -2/8 = -0.25

A sample of n = 20 scores has a mean of M = 45 and a standard deviation of s = 8. In this sample, what is the z-score corresponding to X = 57?​

Z = (x-M)/s = (57-45)/8 = 12/8 = 1.5

For a sample with s = 12, a score of X = 73 corresponds to z = 1.00. What is the sample mean?​

Z = (x-M)/s Transform to compute sample mean M= x - zs = 73 - (1*12) = 61

For a sample with M = 80, a score of X = 88 corresponds to z = 2.00. What is the sample standard deviation?​

Z = (x-M)/s Transform to compute standard deviation S = (x-M)/z = (88-80)/2 = 4

A sample has M = 72 and s = 4. In this sample, what is the X value corresponding to z = -2.00?​

Z = (x-M)/s Transforming to find X X = M+zs = 72 + (-2*4) = 72 - 8 = 64