Draw and thoroughly explain all of the components in a normal distribution bell curve including, the number of cases within 1 and 2 standard deviations from the mean, the areas representing the...

Draw and thoroughly explain all of the components in a normal distribution bell curve including, the number of cases within 1 and 2 standard deviations from the mean, the areas representing the rejection and acceptance of the null hypothesis, and where particular alpha levels lie on the graph.

Expert Answers
steveschoen eNotes educator| Certified Educator

The bell curve essentially "sums up" all the possibilities that can occur given a particular experiment.  Especially, in many cases, as the number of times the experiment is repeated, and all results recorded on the same histogram, the histogram essentially starts to look like a normal/bell curve.

The one thing with the bell curve, the y-axis isn't normally referred to.  We really only refer to the x axis and the area under the bell curve.  The x axis is what we measure from the experiment (the y axis would be how many times we "obtain" that measure).  The "entire area" under the curve is considered 100%, as in the probability of all possibilities occurring is 100%.  So, then, from the middle of the curve to the left or right, each side would be considered 50%, as in 50% of the time we could get lower than the middle value, 50% of the time we could get higher than the middle value.  The middle value in this case would be the "mean value" of your data points.

Now, what has been found is, if you go out to 1 standard deviation (I will use "sd" for standard deviation), so, from the mean value to +1 standard deviation value, approximately 34% of the data values would fall inside that range, or the probability that a data point would fall inside this range is 34%.  So, from -1 sd to +1 sd, the probability would be 68% (34% on one side, 34% on the other side).

Or, using more specific values, let's take this example.  The mean value on an exam is 100 points with a sd of 15 points (like in the attachment).  So, we can draw a normal curve with 100 in the middle, as the mean value, and standard deviations marked in 15 points increments from there.  So, one could consider this 2 fold:

- 68% of the students scored between 85 and 115 points, and/or
- the probability of a student scoring between 85 and 115 points is 68%.

Similarly, with +/- 2 sd, or from 70 to 130 points:

- 95% of the students scored between 70 and 130 points, and/or
- the probability of a student scoring between 70 and 130 points is 95%.

The x axis can be tweeked.  For instance, if you were to change the values on the x axis with the formula:

x new = (x old   -    mean)/sd

The values of the "1 sd", "2 sd", etc., become "1" and "2".  For instance, on our example, the values of 85, 100, and 115, become "-1", "0", and "+1", similarly with all the other values.

As for referring this to the null hypothesis and rejection, it would depend upon the example.  Taking a different example, let's say that 80% of the students passed the exam, so mean = 0.8.  A student group surveyed 90 students and found only 61 students passed the exam.  Does this result suggest the teacher is reporting an incorrect number of students passing the exam?

The null hypothesis would/could be "N = 0.8 (80%)", what is the "believed" value, or "perceived" value.  And, the alternative hypothesis would/could be "N < 0.8", or what's being tested.

Then, from there, there can still be a variety of tests that can be taken.  For instance, we could take a P-test.  First, if 61 out of 90 students, they found that 0.678 (or 67.8% of students passed the exam).  You could mark this on your chart, with 0.8 as the mean and 0.042 as the sd.  The P-value you are finding is the area under the normal curve to the left of 0.678.  Here, it would be a out 0.002, or 0.2% probability.

Here, since we found that the student group's number is consider so far off of the reported values, we would reject the null hypothesis, that apparently the rate of students passing the exam is less than 80%.

Alpha levels essentially refer to the confidence levels, also.  For instance, if you wanted to confirm the %-age of students passing with a 99% confidence level (alpha = 1%), you would make a 99% confidence level and see if your P-value is less than that.  As, in our case, we found that the probability from the student group was 0.2%.  This fell well out of the range of the 99% confidence level and, thus, we would reject the null hypothesis.  If the value fell within the 99% confidence interval, then we would accept the null hypothesis.

As a caveat to all of this, there are many more examples and much more to consider with all of this.  For example, if you are concerned with alpha = 1%, or a 99% confidence level, and it's a 2-tail test, then you would have 0.5% on each side, as oppose to a 1-tail test where you would have 1% marked on one side.  There can be examples of the mean that a student group finds is greater than, less than, or simply "not equal to".  The general techniques are all the same.  However, there can be tweeks in each case.

I hope this helps.

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