# Predicted values and Prediction Intervals

Let the variable $x$ represents the age of a child and the variable $y$ represents the number of days a child absent from school in year 2000. A sample of 6 children give the following data:

$$\begin{array}{|c|c|c|c|} \hline Age, x\hspace{2mm}(years) & 6 & 8 & 9 & 10 & 12 & 14 \\ \hline Number\hspace{2mm}of\hspace{2mm}days, y & 2 & 1 & 3& 4& 6& 5 \\ \hline \end{array}$$

At $\alpha=0.05$, there is enough evidence to conclude that there is a significant linear relationship between the age of a child and the number of days the child absent from school in year 2000. I want to find the predicted number of days a child of 11 years old absent from school in year 2000. Also, I want to find the 95% prediction interval of the number of days a child of 11 years old absent from school in year 2000 using the formula $$\hat{y}\pm{t_{\alpha/2}s_{est}\sqrt{1+1/n+\frac{n(x-\bar{x})^2}{n\sum{x^2}-(\sum{x})^2}}},$$ where $s_{est}$ is the standard of the estimate, $n$ is the sample size, $\bar{x}$ is the sample mean. From a series of calculations, the regression line is $$\hat{y}=-1.918+0.551x.$$ Also, $\bar{x}=9.8$, $\sum{x}=59$, $\sum{x^2}$=621, $t_{\alpha/2}=2.776$, $s_{est}=1.129$ and the degrees of freedom are 4. Now, I want to find the predicted number of days a child of 11 years old absent from school in year 2000. When $x=11$ years, we have $$\hat{y}=-1.918+0.551x=4.143.$$ The variable y is a discrete variable and according to the previous calculation, I want to report the predicted number of days a child of 11 years old absent from school in year 2000. Which answer is the best one?

i) Therefore, rounded to the nearest whole number, the predicted number of days for a child of 11 years old absent from school in year 2000 is 4.

ii) The predicted number of days for a child of 11 years old absent from school in year 2000 is approximately 4.

iii) The predicted number of days for a child of 11 years old absent from school in year 2000 is 4.143.

Any other suggestions? y is a discrete variable. Which value for $\hat{y}$ should I use in the formula to calculate the 95% prediction interval of the number of days a child of 11 years old absent from school in year 2000? $\hat{y}=4.143$ or $\hat{y}=4$? Also, If I use $\hat{y}=4.143$ to calculate the prediction interval, after a series of calculations, I get the answer '0.707 to 7.579'. y is a discrete variable, should I report the rounded answer '1 to 8' instead of '0.707 to 7.579'? If I use $\hat{y}=4$ to calculate the prediction interval, after a series of calculations, I get the answer '0.564 to 7.436'. y is a discrete variable, should I report the rounded answer '1 to 7' instead of '0.564 to 7.436'?

• One reason why you're stuck is that in the middle of your calculations you observe the model is not appropriate for the data. So, rather than assuming the residuals are Normally distributed--which does not accommodate the discreteness of the response very well--why don't you make a more realistic assumption using (say) a generalized linear model?
– whuber
Commented Nov 8, 2014 at 18:40