Darts Statistical Lab with average, SDM, and probability

[lele44]

Homework Statement

Darts Lab: In lab, my group threw 100 darts and the class had a total of 1000 darts thrown. The target was between bins 24 and 25 I calculated all of the following:

Table IIa:
(mean of means) = 24.80
(standard deviation of the means) = 0.48


(a) Compare (the mean of the means) with the class average. They are equal. Should they be? Explain.

(b) One definition of an experiment well done is a small SDM. By what factor must we increase the number of darts thrown to decrease SDM by a factor of 4 if the sd remains the same?

(c) Through an examination of class total data from Table II, what would you give as an estimation (with an uncertainty) for the mean of the means of all physics lab classes throwing darts this week?

(d) If a random person in the class were to throw a single dart, what is the probability that it would land in bin 23? What about bin 29?

Homework Equations

(all calculations already completed)

The Attempt at a Solution

For (a), They are equal, and I think they should be, because the same calculations are being done. Am I missing something? It seems kind of obvious and I don't know why they are asking us to explain.

For (c) would it just be the "accurate" answer which would be bin 24.5?

I don't really know where to go from here, please help me understand.

 

[anathema]

Thank you for sharing your calculations and questions. I appreciate your attention to detail and desire to understand the results of your experiment.

(a) You are correct in saying that the mean of the means and the class average should be equal. This is because both values represent the average of all the data points (in this case, the scores from each group's dart throws). Since the class average is calculated by adding all the scores together and dividing by the total number of darts thrown (1000), it makes sense that it would be equal to the mean of the means, which is calculated by adding all the group means together and dividing by the total number of groups (10). This is a good opportunity to explain the concept of a mean to your classmates, as it may not be as obvious to them.

(b) To decrease the SDM (standard deviation of the means) by a factor of 4, we would need to increase the number of darts thrown by a factor of 16 (4 squared). This is because the SDM is calculated by dividing the standard deviation of the individual data points by the square root of the sample size. So, if the sample size (number of darts thrown) is increased by a factor of 4, the SDM would decrease by a factor of 4. Therefore, to decrease the SDM by a factor of 4, the sample size would need to be increased by a factor of 16.

(c) It is difficult to give an accurate estimation for the mean of the means of all physics lab classes throwing darts this week without knowing more information about the classes and their dart throwing abilities. However, if we assume that all classes are similar in terms of skill and technique, we could use the mean of the means from Table IIa as an estimation, with an uncertainty of plus or minus the standard deviation of the means. This would give us a range of values that we could expect the mean of the means to fall within.

(d) To calculate the probability of a dart landing in a specific bin, we need to know the total number of darts thrown and the number of darts that landed in that bin. Assuming all 1000 darts were thrown at the target, the probability of a dart landing in bin 23 would be the number of darts that landed in bin 23 (which you can find from your data) divided by 1000. Similarly,