Testing Normal Models Using Chi-Square Goodness-of-Fit

Apply the chi-square goodness-of-fit test to assess whether sample data come from a normal distribution: bin the data, compute expected frequencies from the normal CDF, calculate the chi-square statistic, determine degrees of freedom (adjusting when parameters are estimated), and decide whether to reject the null hypothesis.

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Expected Frequencies Under a Normal Model

The chi-square goodness-of-fit test can be used to test whether a sample $X_1, \ldots, X_n$ comes from a specified normal distribution $N(\mu, \sigma^2)$ . Since the normal distribution is continuous, we first partition the real line into $k$ non-overlapping bins $B_1, \ldots, B_k$ and convert the data into bin counts.

Under $H_0$ , the probability that an observation falls in bin $B_i = (a_i, b_i]$ is

p_i = \Phi\!\left(\dfrac{b_i - \mu}{\sigma}\right) - \Phi\!\left(\dfrac{a_i - \mu}{\sigma}\right),

where $\Phi$ is the standard normal CDF. The expected count in bin $B_i$ is

E_i = n \cdot p_i.

For instance, suppose $n = 100$ and $H_0: X \sim N(50, 10^2)$ . For the bin $(40, 60]$ ,

p = \Phi(1) - \Phi(-1) \approx 0.8413 - 0.1587 = 0.6826,

so $E = 100 \cdot 0.6826 = 68.26$ .