What model should I use to prove statistical significance? I'm wondering what method should I use to prove that the trend below is statistically significant?
I want to prove that employees are less satisfied with an event, the longer they have been with the company.
Here is my data:

Thank you for your thoughts in advance!
 A: Granted you shouldn't take any important decision based on this answer, I would
start by simply looking at the data to get a feel for how the correlation between satisfaction and time looks like.
Just by reading off the numbers from your table, this what I get:
Satisfaction <- c(55, 34, 24, 12, 17, 10, 14)
Tenure <- c(0.5, 2, 4.5, 8, 13, 17.5, 20)

par(mfrow= c(1, 2))
plot(Tenure, Satisfaction, ylim= c(0, max(Satisfaction)), pch= 19)
plot(Tenure, Satisfaction, ylim= c(0, max(Satisfaction)), pch= 19, log= 'x', xlab= 'log10(Tenure)')


The correlation after log-trasforming tenure (plot on the right) seems quite linear and convincing.
I think it's reasonable to log-transform time since the difference between,
say, the first and second year is more important than the difference 10th and
11th year.
We can estimate the Pearson correlation coefficient to quantify the linear trend we see. The correlation seems strongly negative (> -0.9) and there is
~0.0005 chance we would see it if the two variables had correlation 0:
cor.test(log10(Tenure), Satisfaction)

# Pearson's product-moment correlation
# 
# data:  log10(Tenure) and Satisfaction
# t = -8.053, df = 5, p-value = 0.0004779
# alternative hypothesis: true correlation is not equal to 0
# 95 percent confidence interval:
#  -0.9947831 -0.7670963
# sample estimates:
#        cor 
# -0.9635451

Again, this is very rough but maybe a simple correlation coefficient suffices for your purpose. (I deliberately avoided concluding that results are/are not statistically significant since I think that concept is misleading and may be better avoided altogether)

EDIT: A few warnings in case this answer gets overinterpreted:

*

*There are only 7 datapoints and each of them is a mean with unknown variance. This adds uncertainty that is not captured by the correlation coefficient as shown above. Also, we need to transform the data in order to get a linear trend so that p-value is not correctly computed.


*The data is binned and especially for the last bin there is some wobble about which value to use as reference.


*Maybe worth repeating the mantra that correlation does not imply causation. Even if that trend is confirmed by more data, you cannot say anything about time of employment affecting satisfaction. Presumably, people in the first bins are very different from those in the last one in terms of age, social status etc. (Besides, how well something as complex as "satisfaction" can be captured by a single metric?) This is just in case someone takes away the message that firing people after three years improves overall satisfaction...! I strongly doubt that!
