1. These probabilities are the probabilities that individual values in a sample will fall in a 50 gram range, and thus represent the integral of individual probability over the range.

2. Probability distributions with a shape similar to the above example are quite common. They are nearly symmetrical and values near the average are more probable than those further from the average.

3. Distributions of this type are often referred to as Gaussian or normal distributions.
Population
And we can estimate the probability distribution of the parent population from statistical estimates of the mean and variance.
Statistic

4. The probability of occurrence of specific values in a sample often takes on a bell-shaped appearance as in the case of our pebble mass distribution.

5. The Gaussian (normal) distribution of pebble masses looks a bit different from the probability distribution we derived directly from the sample.

6. In the probability histogram, each bar represents a discrete sum of masses over a 50 gram range divided by the total number of the pebbles.

7. What do the areas in the above table represent?

10. Consider question 7.4 (page 124) of Waltham - In this question Waltham evaluates the equivalent probability that a pebble having a mass somewhere between
401 (i.e. >400) and 450 grams would be drawn from a normal distribution having the same mean and standard deviation as the sample.
Not from the sample distribution – we already know what that probability is.

11. Note that 401grams lies (401-350)/48 (i. e. +1
Note that 401grams lies ( )/48 (i.e ) standard deviations from the mean.
450 grams lies ( )/48 or standard deviations from the mean value.
1.06 and 2.08 are z-values or standard normal representations of the pebble masses associated with this sample.
Note that Waltham rounds off the mean and standard deviation to 350 and 48, respectively.

12. How can we estimate the area between
p (z = 1.06) and p (z=2.08)?
Note that this area corresponds to the probability that a pebble drawn at random from this ideal parent population will have a value somewhere between 401 and 450 grams.

13. The areas we get from tables
The area we want to find
This Area
The areas we get from tables
Note that we can express that area as one half the difference of the two areas shown above

14. ... one half the combined areas.
yields - X

15. As an example, from our table, we have the areas between  1 and 2 standard deviations from the mean.

16. Areas taken from the table

17. = 0.271
This difference equals the sum of two remaining areas: one between -1 and -2 standard deviations from the mean and the other between +1 and +2 standard deviations between the mean.

18. We’re only after the area on the positive side of the bell between 1 and 2 standard deviations - so take 1/2 the difference. ...Of course, it will work for either side.

19. Confirm for yourself that the area out to + and - 2.08 is 0.962.
Question 7.4 is a little more complicated. We no longer have numbers listed in the table.
Waltham goes through a weighted average determination of the area under the curve between + and standard deviations. He obtains the area 0.71.
Confirm for yourself that the area out to + and is
The difference is
Now we take one-half of that to get 0.126

20. This method of linear interpolation assumes linearity in the curve between 1 and 1.1
P(1.06) is six-tenths of the way from P(1) and P(1.1) or plus 0.6 times the difference (0.046)
= = 0.71

21. P(2)=0.954
P(2.1)=0.964
P(2.08) is eight-tenths of the way from P(2) and P(2.1) or plus 0.8 times the difference (0.01)
= = 0.962

22. Be sure you understand the difference
=0.252
One half of that =
0.126 is the normal probability of obtaining a pebble with mass between 401 and 450 grams from the beach under investigation.
Note that the value derived from the normal distribution compares nicely with that observed in the sample (0.126 vs. 0.14).
Be sure you understand the difference

23. Be sure to finish reading Chapter 7
Be sure to finish reading Chapter 7. On Tuesday we will consider the idea of statistical difference. Look at problem We will discuss this in class on Tuesday as well.
Hand in the PsiPlot histogram and probability distribution. Your effort will be graded so be sure to put you name on the plot

24. We may begin discussion of derivatives next Tuesday
Start Reading Chapter 8
We may begin discussion of derivatives next Tuesday

25. Bring Up Excel