We routinely use the normal quantile plot to check for normality. One can also use a chi-squared test. For that, we have to group the data into bins, converting numerical data into categorical data.
The following figure shows the normal quantile plot of daily stock returns in 2010 on the value-weighted total U.S. market index.

The following table counts the number of returns falling into 8 intervals. The table includes the count expected under the assumption that these data are normally distributed, using the sample mean x = 0.0009874 with SD = 0.0151.

(a) What does the normal quantile plot indicate about the distribution of returns?
(b) The table groups all returns that are less than -0.03 and more than 0.03. Why not use more categories to separate very high or low returns?
(c) Compute the chi-squared test of goodness of ft and its p-value, noting that we have to estimate two parameters from the data in order to find the expected counts.
(d) Does the chi-squared test agree with the normal quantile plot?
(e) What€™s the advantage of using a normal quantile plot to check for normality? The advantage of using the chi-squared test?

0.07 0.06 0.05 0.04 0.03 0.02 0.01 -2.33-1.64-128-0.67 00 0.67 1.28 164 2.33 -0.01 0.02 0.03 0.04 0.05 50 100 0.05 0.2 0.5 0.8 0.95 Normal Quantile Plot Count Range Count Expected Count 18 19 56 128 178 10.02 31.24 76.16 121.42 126.61 86.37 38.53 13.66 -0.03 -0.03-0.02 -0.02x s-0.01 0 < s0.01 0.01 x0.02 0.02 < x -0.03 0.03