Statistical tests Some students were asked how they prefer be informed.
The results of the research are below:
television: 22
radio: 15
newspaper: 26
internet: 15
The researchers want to know if the students prefer a specific way to be informed or not.
Which statistical test should i use and how?
 A: If your goal is to determine whether the three methods are equally
popular, or confirm to some pre-specified distribution on the four
methods, then a chi-squared test is appropriate.
It must be true that $22 + 15 + 26 + 15 = 78 = n$ subjects each
chose exactly one method as favorite.
We test the null hypothesis is $H_0: p_T = p_R = p_N = p_I = 1/4$
against the alternative than not all methods are equally popular.
Then, under $H_0,$ we expect $E = 78/4 = 19.5$ responses favoring
each method. The question is whether the observed values
$X = (22, 15, 26, 15)$ are far enough from 19.5 each to reject $H_0.$
Under $H_0,$ the test statistic $Q = \sum_{i=1}^4 \frac{(X_i = E)^2}{E}$
has approximately a chi-squared distribution with $k - 1 = 3$ degrees
of freedom.  For our data $Q = 4.54.$
X = c(22, 15, 26, 15)
Q = sum((X - 19.6)^2/19.6);  Q
## 4.542857

The PDF of $Chisq(\nu = 3)$ is shown below. The vertical blue line
is at 4.52. The area under the curve to the right of that line is
the P-value = 21.2% of the test.

If the P-value were below 5% we
would reject $H_0.$ (Put in another way, the 'critical value' for
the test is 7.8147; we would not reject $H_0$ unless $Q \ge 7.8147.$
The critical value cuts 5% from the right tail of $Chisq(3); it can
be found in printed tables of the chi-squared distribution.)
Although the observed values $X = (22, 15, 26, 15)$ may not seem
to be 'close' to 19's or 20's, they are close enough not to be
significant evidence against $H_0.$
Cautionary Note: If we had $n = 312$ and observed preference counts
$X = (88,  60, 104,  60)$ then proportions in the four categories
would be exactly the same as for your data (all counts just multiplied by 4).
But then $Q = 176.6$ and $Q$ is still compared with the distribution $Chisq(3).$
The conclusion would be overwhelmingly in favor or rejecting $H_0.$
It is neither relative proportions nor degrees of freedom that tell
the whole story! Sample size $n$ also matters in a way that might not
be immediately obvious.
