# Determine the standard deviation for statistical hypothesis testing when not explicitly given

I have got a video in grayscale, and I am interested to find out if the intensity of a pixel at a given location is more or less constant over time. I have been told here that I can use linear regression first to find the formula $y = at + b$, and then use statistical hypothesis test to find out if the null hypothesis $a = 0$ should be rejected. If it is rejected, then the pixel intensity is not constant over time.

I believe during the test, I can safely assume a normal distribution of the value $a$. However, while the mean is $0$ (according to the null hypothesis), what should be the standard deviation? Is there any general method to determine it, or do I need other knowledge to determine it?

• Do ANOVA on the data and get the standard deviation?stat.yale.edu/Courses/1997-98/101/anovareg.htm – Satish Ramanathan Dec 28 '15 at 10:36
• Thanks for the answer, but as I am new in this area, may I ask more detail about the steps? Assume I have $n$ data points $(t_1, y_1) ... (t_n, y_n)$, I use linear regression to find the formula $y = at + b$, then I shall use ANOVA, right? In that case, is $\^{y_i}$ the value of $y$ according to the equation found, $y_i$ the actual data, and $\bar{y}$ the actual mean of the $y_i$? – GreenPenguin Dec 29 '15 at 4:52
• Look at the different answer that I posted, Hopefully you can follow that procedure for your simple test – Satish Ramanathan Dec 29 '15 at 5:17
• I think I get it. Thanks! – GreenPenguin Dec 29 '15 at 8:55
• You are welcome!! – Satish Ramanathan Dec 29 '15 at 9:42

See the numerical example of Simple Linear regression at this Website (https://en.wikipedia.org/wiki/Simple_linear_regression) and follow exactly the procedure where your t's are their x's and you a in the model $y=at+b$ is their $\beta$. Find the confidence interval of "a"($\beta$) and find if 0 is contained in it. If it contains 0 accept that the hypothesis that pixel intensity is constant over time and if it does not contain 0 then conclude that pixel intensity is not constant over time.