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# Statistical Calculations, Representations and Results

M = Mean
S2 = Population Variance
SM = Standard Deviation of the Distribution of Means
t = score for your sample
t needed = cut-off score that establishes the region of rejection (also known as the critical value)
Decision: Reject the Null or Fail to Reject the Null (select only one)

Evolutionary theories often emphasize that humans have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24-hour cycle of sleeping and waking, even if they are not exposed to the usual pattern of sunlight. To test this notion, eight paid volunteers were placed (individually) in a room in which there was no light from the outside and no clocks or other indications of time. They could turn the lights on and off as they wished. After a month in the room, each individual tended to develop a steady cycle. Their cycles at the end of the study were as follows: 25, 27, 25, 23, 24, 25, 26, and 25.

M =
S2 =
SM =
t =
t needed = +
Decision:

Below is an explanation of the symbols in Chapter 8, Practice Problem 18.

N1 = number of participants in the experimental group
N2 = number of participants in the control group
df1 = degrees of freedom for the experimental group
df2 = degrees of freedom for the control group
dfTotal = degrees of freedom for both groups
M1 = mean of the experimental group
M2 = mean of the control group
S21 = estimated population variance of the experimental group
S22 = estimated population variance of the control group
S2Pooled = pooled estimate of the population variance
S2M1 = variance of the distribution of means for the experimental group
S2M2 = variance of the distribution of means for the control group
S2Difference = variance of the distribution of differences between means
SDifference = standard deviation of the distribution of differences between means
t = score for your sample
t needed = cut-off score that establishes the region of rejection (also known as the critical value)
Decision: Reject the Null or Fail to Reject the Null (select only one)

18. Twenty students randomly assigned to an experimental group receive an instructional program; 30 in a control group do not. After 6 months, both groups are tested on their knowledge. The experimental group has a mean of 38 on the test (with an estimated population standard deviation of 3); the control group has a mean of 35 (with an estimated population standard deviation of 5).

N1 =
N2 =
df1 =
df2 =
dfTotal =
M1 =
M2 =
S21 =
S22 =
S2Pooled =
S2M1 =
S2M2 =
S2Difference =
SDifference =
t =
t needed = +
Decision:

Chapter 9 Instructions
Practice Problem 17

Below is an explanation of the symbols in Chapter 9, Practice Problem 17.

S2Between = between groups population variance estimate
S2Within = within groups population variance estimate
F = statistical score that represents the ratio of the between groups to the within groups population variance estimate
F needed = cut-off score that establishes the region of rejection (also known as the critical value)
both groups
Decision: Reject the Null or Fail to Reject the Null (select only one)

17. Do students at various colleges differ in how sociable they are? Twenty-five students were randomly selected from each of three colleges in a particular region and were asked to report on the amount of time they spent socializing each day with other students. The results for College X was a mean of 5 hours and an estimated population variance of 2 hours; for College Y, M = 4, S2 = 1.5; and for College Z, M = 6, S2 = 2.5.

S2Between =
S2Within =
F =
F needed = +
Decision:

Chapter 11 Instructions
Practice Problem 11 and 12c & 12d

For Chapter 11, Practice Problem 11, follow the instructions below to create 6 scatter plots using Microsoft Excel.

Note: The following instructions may not work with all versions of Excel. If you use Windows Vista, please scroll down to find instructions for creating scatter plots with the Windows Vista platform.

On most versions of Excel, do the following:

Open a blank Excel document. Click on Insert - then Chart. The chart Wizard will appear. Under the Standard Types tab, click on "XY (Scatter)"

Under the Chart Sub-Type: highlight the top box. It should be "Scatter, Compares pairs of values." click next, which should take you to "Chart Wizard - Step 2 of 4.

Click the Series tab. Click the Add button. In the Name Box, type the name of the scatter plot you will be designing, (i.e.) Perfect Positive, Perfect Negative, etc. In the X Values box, type in the values of the X coordinates you wish you use in your scatter plot. Separate numbers by commas. In the Y Values box, type in the values of the y coordinates you wish you use in your scatter plot. Separate numbers by commas. Click Next to go to Step 3 of 4. Do nothing in Step 3.

Click Next to go to Step 4 of 4 - Chart Location: In the Place Chart option, check "As New Sheet" Chart 1 should appear in the white box. Click OK, or Finish.

Click Insert to create the next chart - scatter plot. Your second scatter plot will be Chart 2; your third scatter plot will be Chart 3, etc.

Instructions for Windows Vista users are listed below:

On a new blank excel sheet, place your scores in two columns X and Y. Put these in rows A and B.
Click on the insert tab and in the charts sub-group, click the Scatter option. Click on the "scatter with only markers option."
This will present the blank chart and you are now in the Design mode. Second sub-group from the left is "Data." Click the "select data" button to open the 'select data source.'
Click hold and drag to select all the scores (this will be all the numbers in columns A and B). Click Ok and your scores will be put into the chart. Be sure not to select the "X" and the "Y".
Next to the "design" button is the "Layout" button.
In the layout mode, click on the "Axes" box and you can create the parameters of the horizontal and vertical axes. For example, lay your cursor over the Primary horizontal axes. At the bottom is "More primary horizontal axes options." Click on this to open the window to set your values, i.e. in axis options here you can set you minimum and maximum values as well as your units, e.g. .5 or 1. I found that your scores can be in decimal increments and will show just fine in a chart whose units are whole numbers.
Be sure and get the chart type you want, while in the "design" mode on the far left is "change chart type". Here you can be sure and select the scatter with markers only option. Also in the design mode, is chart layouts. Be sure you have selected the layout you want here as well.
To put chart title and such into your chart go to the lay out or format mode, the two options to the right of the design tab. On the far left is the current selection box. If you click on "chart area" the options for modifying specific descriptions within your chart such as the title, etc.
If you want to include a trendline, this option is in the "layout Mode," in the analysis button.
To change the "series 1" that appears in your chart to match your specific description, you have to go the 'design' mode and go to the 'select data' button. When the 'select data source' opens, click on 'series 1' to highlight and click 'edit' right above it. Here you can put your own info in and it will then appear on your chart. This will also create a chart title that will appear at the top of your chart.

Below is an explanation of the symbols in Chapter 11, Practice Problem 12c & 12d. Note: Instructions for Practice Problem 12c & 12d begin at the top of page 489 in your textbook (above problem 12)

r = score that represents the correlation coefficient
t = score that determines the significance of the correlation coefficient score
t needed = cut-off score that establishes the region of rejection (also known as the critical value)
both groups
Decision: Reject the Null or Fail to Reject the Null (select only one)

11. Make up a scatter diagram with 10 dots for each of the following situations: (a) perfect positive linear correlation, (b) large but not perfect positive linear correlation, (c) small positive linear correlation, (d) large but not perfect negative linear correlation, (e) no correlation, (f) clear curvilinear correlation.

12. Four research participants take a test of manual dexterity (high scores mean better dexterity) and an anxiety test (high scores mean more anxiety). The scores are as follows.
Person Dexterity Anxiety
1 1 10
2 1 8
3 2 4
4 4 -2

For problems 12 ( c) and 12 (d) do the following:
(c) figure the correlation coefficient;
(d) figure whether the correlation is statistically significant (use the .05 significance level, two-tailed)

#### Solution Summary

This solution determines the mean, the population variance, standard deviation, the sample score, the cut-off score, and the decision whether to reject the null hypothesis for a case study looking at sleep patterns.

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