Question

A STATISTIC GENIUS please...
I have the following two problems:
4.27: A set of data whose histogram is bell shaped yields a mean and standard deviation of 50 and 4, respectively. Approximately what proportion of observations
a. are between 46 and 54?
b. are between 42 and 58?
c. are between 38 and 62?

4. 28: Refer to Excercise 4.27. Approximately what proportion of observations
a. are less than 46?
b. are less than 58?
c. are greater than 54?

My textbook tells me that I should use the Empirical Rule. But I don't think the rule makes any sense in this context...

Answer 1

what is does the rule state?

Answer 2

Empirical Rule:
1. Approximately 68% of all observations fall within one standard deviation of the mean.
2. Approximately 95% of all observations fall within two standard deviations of the mean.
3. Approximately 99.7% of all observations fall within three standard deviations of the mean.

Answer 3

ok, now what do you think that rule is saying? how could we apply it to the problem?

Answer 4

Because the histogram is bell-shaped, the Empirical Rule applies. I first thought to do it like this:

1. Approximately 68% of the data lie between 46% (the mean minus one standard deviation = 50 - 4) and 54% (the mean plus one standard deviation = 50 + 4).
Is it like that?

Answer 5

the %-signs are not correct .. but the thought process is fine.

68% of the data is within reach of the mean by 1 standard deviation ... so, between (50-4(1)) and (50+4(1)) we should have approximately 68% of the data.

within reach of 2 standard deviation? our range is calculted as, (50-4(2)) and (50+(4(2)) right?

Answer 6

the 2nd question gets a little tricky, since it suggests you recall symmetry and that all data equals 100% to start off with

Answer 7

So in an exam situation, what should be written? Im sorry, I'm not really understanding

Answer 8

Approximately what proportion of observations:

are between 46 and 54? determine the number of standard deviations each end point is from the mean

\[z=\frac{x-\mu }{\sigma}\]

then construct your answer using the empirical rule as your proof: There is ___% of the data between 46 and 54 since they are within ____ standard deviations of the mean.

Answer 9

conceptually, a normal distribution has a mean of zero, we can move any data set to have a mean of zero: mean - mean = 0

but whatever we do to one point we do to all the point: x- mean = (new x)

a normal distribution has a standard deviation of 1 ... since thats a nice way to count things is 1 at a time. we determine how many times something does into another, by division.

how many standard deviations (sd) are are between x and the mean? (new x)/sd tells us that.