1. What is the probability that a single randomly chosen value (from 1 to 100) would be over 60? [Note that this is a uniform distribution, so don't use the NormalCdf command, think of it like a spinner with 100 slots]

Since the distribution is uniform, each number from 1 to 100 has an equal chance of being chosen. There are 40 numbers from 61 to 100 (inclusive). So, the probability of choosing a number greater than 60 is the number of favorable outcomes (numbers greater than 60) divided by the total number of outcomes (numbers from 1 to 100).

This gives us 40/100 = 0.4 or 40%. So, the probability that a single randomly chosen value from 1 to 100 would be over 60 is 40%.

2. What is the probability that the mean of a group of 5 randomly chosen values (from 1 to 100) would be over 60? The NormalCdf command is now appropriate. [Note the population mean is 50.5 for all the questions on this activity. The SD is 29.01, but when you group of 5 means together, it shrinks the SD down. In this case, these would then have an SD of 29.01/sqrt(5) = 12.97]

The probability that the mean of a group of 5 randomly chosen values (from 1 to 100) would be over 60 is 0.2321 0r 23.21%.

3. Look over the Google doc - what proportion of size 5 samples have a mean over 60? How close did we get to the proportion the mathematics in #2 predicts?

Out of 92 observations from Google doc, there are 25 samples means greater than 60.

Hence, we have a proportion of 25/92 = 0.2717 or 27.17%

Yes, the obtained proportion of 27.17% is close to the probability obtained above 23.21%.

4. What is the probability that the mean of a group of 20 randomly chosen values (from 1 to 100) would be over 60? [Again, when you group means together, it shrinks the SD down. In this case, these would then have an SD of 29.01/sqrt (20) = 6.49]

The mean of a group of 20 randomly chosen values (from 1 to 100) would be over 60 is approximately 0.0721 or 7.21%

5. Look over the Google doc - what proportion of size 20 samples have a mean over 60? How close did we get to the proportion the mathematics in #4 predicts?

Out of 46 observations from Google doc, there are 3 samples means greater than 60.

Hence, we have a proportion of 3/46 = 0.0652 or 6.5217%

Yes, the obtained proportion of 6.5217% is close to the probability obtained above 7.21%.

6. What is the probability that a single randomly chosen value (from 1 to 100) would be between 40 and 55? [Note that this is a uniform distribution, so don't use the NormalCdf command, think of it like a spinner with 100 slots]

The probability of randomly selecting a number between 40 and 55 in this uniform distribution is 16%.

7. What is the probability that the mean of a group of 5 randomly chosen values (from 1 to 100) would be between 40 and 55? [see #2 for the SD]

8. Look over the Google doc- what proportion of your group's samples of 5 have a mean between 40 and 55? How close did we get to the proportion the mathematics in #7 predicts?

9. What is the probability that the mean of a group of 20 randomly chosen values (from 1 to 100) would be between 40 and 55? [see #4 for the SD]

10. Look over the Google doc- what proportion of your group's samples of 20 have a mean between 40 and 55? How close did we get to the proportion the mathematics in #9 predicts?

11. Use the SD you found in (2) that has been adjusted for the sample size, and the Empirical Rule to say what 68% of the sample means will fall between in samples of size 5.

12. Working backwards: With a sample of size 20, what sample mean would fall at the 15th percentile? The 70th percentile?

13. Reflect: Keeping in mind that you're finding the mean of the data you get, why is a sample of 20 any better than a sample of 5? That is, WHY are the size 20 sample means closer to the actual population average when compared to the size 5 means? (note: you answered this question weeks ago at the end of Activity 1, this is chance to go deeper, as you have more insight at this point in the course).

Goog doc link attached below

https://docs.google.com/document/d/11gxObL64et2Tqss0DDY3doicwv_fmOq2Cugw3SUIaMs/edit