Calculate the exact slope m (rather than a decimal approximation) of the straight line through the given pair of points, if defi
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Answer: See the table below (attached image)
- SSb = 2345
- SSt = 5639
- DFb = 5
- DFw = 54
- MSb = 469
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Explanation:
I'll start at the right side of the table and work my way toward the left side.
I recommend you have a one way ANOVA formula sheet handy. Such a formula sheet should be located somewhere in the ANOVA chapter of your stats textbook, or toward the appendix section. There are various free online resources to search out as well.
On that formula sheet you should find that
(MSb)/(MSw) = F
where MSb and MSw are the mean squares between and within respectively.
We can find the value of MSb like so
(MSb)/(MSw) = F
MSb = F*MSw
MSb = 7.682*61
MSb = 468.602
This rounds to 469, so we'll go with MSb = 469. The reason I'm rounding is that I've done a similar problem in the past, and the computer system wanted me to round to the nearest whole number. Also, note how every value of the table given to us consists of whole numbers. Of course, your teacher may want you to keep the decimal value instead. So I would ask for clarification if possible.
Since we're going with MSb = 469, this means 469 goes in the last box of row 1.
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Now go back to your reference sheet to locate this formula
MSw = (SSw)/(DFw)
Rearrange this formula to get
DFw = (SSw)/(MSw)
Now plug in the values we have from the table to find,
DFw = (SSw)/(MSw)
DFw = (3294)/(61)
DFw = 54 which goes in the box found in the second row.
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Now recall that
DFb + DFw = df
where DFb is the degrees of freedom between and df is the total degrees of freedom, which in this case is 59.
Using the table and the value we found earlier, we can see that,
DFb = df - DFw
DFb = 59 - 54
DFb = 5 this goes in the second box of row one (ie above the 54).
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Also on your formula sheet, you should see something like
MSb = (SSb)/(DFb)
where each lowercase 'b' stands for "between", and the other abbreviations are what were discussed earlier.
That formula rearranges into
SSb = (DFb)*(MSb)
Then we can use the values we calculated earlier to find
SSb = (DFb)*(MSb)
SSb = (5)*(469)
SSb = 2345 this goes in the first box of row one.
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Lastly, computing the total sum of squares (SSt) will simply involve us adding the two SSb and SSw values to get
SSt = SSb + SSw
SSt = 2345 + 3294
SSt = 5639 this goes in the box of row three.
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To summarize, we have these values to fill into the table
- SSb = 2345
- SSt = 5639
- DFb = 5
- DFw = 54
- MSb = 469
where we have these abbreviations
- SS = sum of squares
- MS = mean square
- DF = degrees of freedom
- b = between
- w = within
Check out the diagram below for the filled in values.
To summarize the formulas used, we basically have the template of
MS = (SS)/(DF)
for both between and within. And we also have the formula F = (MSb)/(MSw). So effectively, the values are found through various division formulas or rearrangements of such. I'm referring to working along any given row. For any given column, we add up the values to get the totals, though that of course does not apply to MS or F.
Admittedly, ANOVA tables and concepts can be confusing because there are a lot of variables and steps going on. But once you get used to it, it's not too bad if you can see how the patterns are emerging (the division and additions I was referring to earlier).
Answer:
. The closest value from your given choices is C)r=0.69
Step-by-step explanation:
To solve this we are using the standard exponential growth equation:
where
is the final population after
years
is the initial population
is the growth rate in decimal form
is the time in years
We know from our problem that the initial population is 5000, the final population is 10000, and the time is 10 years, so ,
, and
.
Let's replace the values and solve for :
Divide both sides by 5000
Take root of 10 to both sides
Subtract 1 from both sides
We can conclude that the growth rate of our exponential equation is . The closest value from your given choices is C)r=0.69