Question

Problem:

To study the effectiveness of possible treatments for insomnia, a sleep researcher conducted a study in which four participants were instructed to count sheep (the Sheep Condition), four were told to concentrate on their breathing (the Breathing Condition), and four were not given any special instructions. The average number of minutes taken for each participant to fall asleep over the next seven days were 14, 28, 27, and 31 minutes for the Sheep condition; 25, 22, 17, and 14 minutes for those in the Breathing condition; and 45, 33, 30, and 41 for the Control condition. Using the .05 significance level, did the different techniques have different effects?

Step 1: Restate the question as a research hypothesis and a null hypothesis about the populations.

            Pop 1: ???

            Pop 2: ???

            Pop 3: ???

Null hypothesis: ???

Research hypothesis: ???

Step 2: Determine the characteristics of the comparison distribution.

What type of distribution is this???

What are the df for the between-groups variance estimate: ???

What are the df for the within-groups variance estimate: ???

Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.

Use the F table (table A-3) in the text book to look up the cutoff score using the degrees of freedom we just stated in Step 2. Cutoff score = ???

Step 4: Determine your sample’s score on the comparison distribution.

This is where you determine the sample’s F ratio (the numerator and denominator). To do that, you have to find the between-groups variance estimate and the within-groups variance estimate.

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Within-groups variance estimate: This is the number that goes on the bottom of your F ratio.

I have a table started to help you come up with the numbers you need. I have filled in the given numbers. You must fill in the blanks, add things up, etc., according to what you learned from the chapter.

Sheep Condition                     Deviation from Mean              Squared Dev from Mean

14

28

27

31

Mean of Sheep Condition People = ???

S₂ = ???

Breathing Condition                Deviation from Mean              Squared Dev from Mean

25

22

17

14

Mean of Breathing Condition People = ???

S₂ = ???

Control Condition                    Deviation from Mean              Squared Dev from Mean

45

33

30

41

Mean of Control Condition People = ???

S₂ = ???

Now you have to use the numbers you just came up with to plug into the formula for the within-groups variance estimate (given below). The “answer” you get is your final number for the within-groups variance.

S²_Within= S²₁ + S²₂ + S²₃ / N_Groups

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Between-groups variance estimate: This is the number that goes on the top of your F ratio.

To get this number, you have to do a few more calculations. You will use the formulas from the book. Use the table below to help get you started (where I inserted the names of groups in parentheses is where you actually input the mean of each group and delete the group names I inserted—I just put them there to guide you).

What is the value of the Grand Mean: ???

Sample Means                        Deviation from Grand Mean (GM)                 Sq Dev from GM

(sheep mean)                         ?                                                                      ?                     

(breathing mean)                    ?                                                                      ?                     

(control mean)                        ?                                                                      ?                                             

Now you have to use the formula for the between-groups variance estimate (given below).

S²_Between = S²_M (n)

To find the S²_M, you have to use the calculations from above with the squared deviation scores from the Grand Mean. Look in your text at the examples if you need help plugging numbers in. Your quick hint is to fill out the table I made you by subtracting the GM from each sample mean, then squaring those answers, adding them all up, and then dividing them by the df_Between. The df_Between is simply the number of groups minus 1. Once you have the S²_M, don’t forget the last step of multiplying it by ‘n’, which is the number of participants in each sample (in this case, that number is 4).

I hope this helps you get the correct numbers to plug into your F ratio!

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What is the F ratio??? ____ / ____ = ______

When you divide out the F ratio, what is your final F value?

Step 5: Decision

Is your F value extreme enough to reject the null hypothesis? What is your final decision and why are you making this decision?

Answer 1

	Sheep Conditon	Deviaiton from mea	Sqared dev from mean
	14	-11	121
	28	3	9
	27	2	4
	31	6	36
Mean	25

Sum of square = 170

As per the sheep condition, we have calculated the mean and respective table

	Breath Conditon	Deviaiton from mea	Sqared dev from mean
	25	5.5	30.25
	22	2.5	6.25
	17	-2.5	6.25
	14	-5.5	30.25
Mean	19.5

Sum of square = 73

As per the Breath condition, we have calculated the mean and respective table

	control Conditon	Deviaiton from mea	Sqared dev from mean
	45	7.75	60.0625
	33	-4.25	18.0625
	30	-7.25	52.5625
	41	3.75	14.0625
Mean	37.25

Sum of square = 144.75

As per the control condition, we have calculated the mean and respective table

Sum of square withing = 170+73+144.75 /3 = 129.25

Grand mean of all the observation is 27.25

	Mean	Deviaction from grand mean	sq dev from grand mean
Sheep mean	25	-2.25	5.0625
Breathin mean	19.5	-7.75	60.0625
Control mean	37.25	10	100
Grand mean	27.25

Sum of square between group is 165.125 * 4 = 660.5

now

for within the tabel we have degree of freedom = 3* (4-1) = 9

For between the gorup wehave degree of freedom = 3-1 =2

So F value = ( Sum of square variance of betweein group / df of between group) /( Sum of square variance of within group / df of within group)

F = 660.5/2 / 387.75/9

F = 7.6653

Now if we look the F value at the 2 df of numerator and 9 df of denomentator we will have 4.2564

So calculated F value is higher than the critical F value so we can reject the null hypothesis and accept the alternate hypothesis.

And state that, there is a difference in group of Sheep, Breath and control group.