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Mathematics & Statistic Tutor Perth - SPSS Help   

 Chi-Square Test of Independence  
 Tests of Significance   Normal distribution   
 SPSS  Instructions   

(a)  Write down the linear regression equation.

(b) What is the value of the standard error of the estimate?

(c) How many degrees of freedom are associated with the t-value for the line of regression?

(d) What is the value of the correlation coefficient?

(e)  Confidence and Prediction Interval

(f) What is the 95% confidence interval for the mean value of   Ŷ when x =   ?

(g) What is the 95% prediction interval for    Ŷ   when x =   ?

What I have tried to do here is put as simply as possible how to answer a variety of questions using SPSS output. The following tables are an example of the output and then I have shown where the information is to answer certain questions. 

Coefficients

Model Summary

ANOVA

(a)  Write down the linear  regression equation.

 Ŷ  = b₀ + b₁x = 2.129 +  .338x

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(b) What is the value of the  standard error of the estimate?

This has another name the standard deviation of y about the regression line.  It tells us how much the  observed values differ from the values on the regression line.  It gives us an  idea of the scatter of the points around the line of regression.

In the formulae for the prediction interval concerning regression, this value is represented by the letter, s

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(c) How many degrees of freedom  are associated with the t-value for the line of regression?

This is (n - 2) degrees of  freedom and is given in the analysis of variance.

So even without knowing  the sample size, you can find the degrees of freedom by looking on the Residual  line in the ANOVA. You can, also, state the size of the sample which in this case is 8

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(d) What is the value of the  correlation coefficient?

The correlation  coefficient is given by R and is a measure of the  linear association between the variables.

The coefficient of  determination, R  Square, gives an  indication of how good a choice the x-value (independent variable) is in  predicting the y-value (dependent variable). It describes the amount of  variation in y-values explained by the regression line.  The larger the value  the better the regression line describes the data.

You would have to look  ‘sign of  b₁’,  to check to see if it was positive or negative correlation.

You do not have to worry  too much about the adjusted value of R Square as it just takes into account the  actual sample size.

Durbin-Watson should be between 1.5 and 2.5  indicating the values are independent.

So the value of the  correlation coefficient is 0.941.

So the value of the  coefficient of determination is 0.886

So the value of the  Durbin - Watson is 2.321

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(e)  Confidence and Prediction  Intervals

We now have to realise that the predicted value can be viewed in two ways:

The mean response of all y  values for a particular value of x.

This is written as  

2.  The  predicted value of y for an individual at a particular value of  x.

This is written  as Ŷ 

It is still the same number just different symbols!!

This information is given in the data  window and not the output window.

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(f) What is the 95%  confidence interval for the mean value of  Ŷ   when x =  ?

The confidence interval  for the population mean of all y values associated with the value of x  chosen.

It is similar to all other  confidence intervals in its interpretation and it involves a t-value and  standard error (sample standard error of predicted value).

This confidence interval  is found under the headings “lmci_1” and “umci_1” in the data window. This would  give the upper and lower limits of the confidence interval so in an examination,  it will probably be a different level of confidence i.e. 90% rather than the  output’s 95%. This means that you will look up the 90% t-value in tables. All  the other values remain the same so can be found in the output.

This value,   is found in the output under the heading “sep_1”
“sep_1” is the sample standard error of the predicted value.

The value,    is found in the output  under the heading “pre_1”
“pre_1”  is the predicted value.

The “_1” is just in case  you have more than one confidence or prediction interval so they can be  distinguished from each other.

You may be given the value  of    in the output,  you can find    by the regression line or under the heading  “pre_1” and then substitute the values into    along with the  t-value.

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(g) What is the 95%  prediction interval for   Ŷ  when x =  ?

The prediction interval is  an interval of a prediction for an individual for a certain value of x.  This is  more variable than the confidence interval as we are dealing with individuals  rather than averages. Here is the formula and you can see the extra term   ‘s²’

where 

The “ s ” is  standing for the sample standard error of the estimate

 The  “ ” is standing for the sample  standard error of the predicted  value

This prediction interval  is found under the headings “lici_1” and  “uici_1”  in the data window.  This would  give the upper and lower limits of the prediction interval so in an examination,  it will probably be a different level of prediction i.e. 90%  rather than the default output 95%. Once again look up the 90%  t-value and substitute in with all the  other values from the output.

In the output, the symbol  “i” is used to denote individual.

The value,    is found in the output  under the heading “sep_1”
“sep_1” is the sample Standard error of the predicted value.

The value,  Ŷ  is found in the output under the heading ”pre_1”
“pre_1” is the  predicted value.

The value, s, is found in  the table below:

Once again you will just  have to substitute the values into the equation

The values of  Ŷ  and  are different ways of expressing the predicted  value.

I hope you followed this  quick explanation of the SPSS output for regression.

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