Student's t-Distribution

Study guide: Exam MAS-I

Student's t-Distribution $t(n)$ with $n$ degrees of freedom is the distribution of a standard normal variable divided by the square root of an independent chi-squared variable over its degrees of freedom. It is the sampling distribution of a mean or a regression coefficient standardised by an estimated standard error.

$T = \frac{Z}{\sqrt{V/n}} \sim t(n), \quad Z \sim N(0,1),\ V \sim \chi^2(n)$

$f(t) = \frac{\Gamma\!\left(\frac{n + 1}{2}\right)}{\sqrt{n\pi}\;\Gamma\!\left(\frac{n}{2}\right)}\left(1 + \frac{t^2}{n}\right)^{-(n + 1)/2}$

$E[T] = 0, \quad n > 1$

$\text{Var}(T) = \frac{n}{n - 2}, \quad n > 2$

$Z$ and $V$ are independent, and $f$ is defined for all real $t$ . The density is symmetric about 0 with heavier tails than the normal; as $n \to \infty$ it converges to the standard normal density. With $n = 1$ it is the Cauchy distribution, which has no mean.
Normal samples: for a sample of size $n$ from $N(\mu, \sigma^2)$ with sample mean $\bar{X}$ and sample variance $S^2$ , $\dfrac{\bar{X} - \mu}{S/\sqrt{n}} \sim t(n - 1)$ . This is the one-sample t-test and the confidence interval for $\mu$ when $\sigma$ is unknown.
Regression coefficients: in simple linear regression, $t = \hat{\beta}_1/\text{SE}(\hat{\beta}_1)$ has a $t(n - 2)$ distribution when $\beta_1 = 0$ ; a 95% interval for $\beta_1$ uses the 97.5% quantile of $t(n - 2)$ .
Link to F: $T^2 \sim F(1, n)$ , so a two-sided t-test on one coefficient is the F-test with one numerator degree of freedom.
The CAS MAS-I table gives two-sided tail areas $0.20,\ 0.10,\ 0.05,\ 0.02,\ 0.01$ : the entry $c$ satisfies $P(|T| > c) =$ the column heading. It covers $df = 1$ to 30, then 35 to 60 in fives, 70, 80, 90, 100, 120, 400 and $\infty$ (the normal row). For a one-sided test at level $\alpha$ , read the $2\alpha$ column.
On MAS-I it supplies critical values for tests of means and for the t-statistics reported for linear-model coefficients.

Has Average Severity Risen? Last year's average claim severity was 1,000. A sample of 16 claims this year has $\bar{x} = 1{,}150$ and $s = 200$ ; claim sizes are assumed normal. Test $H_0: \mu = 1{,}000$ against $H_1: \mu > 1{,}000$ at the 5% level, and bound the p-value.
Answer
 $\begin{align*} t &= \frac{1{,}150 - 1{,}000}{200/\sqrt{16}} \\ &= \frac{150}{50} \\ &= 3.0 \end{align*}$ 
There are $16 - 1 = 15$ degrees of freedom. The test is one-sided, so the 5% critical value is in the two-sided $0.10$ column: $1.753$ . Since $3.0 > 1.753$ , reject $H_0$ . The two-sided $0.01$ column gives $2.947 < 3.0$ , so the one-sided p-value is below $0.005$ .

Significance of a Rating Factor in a Regression A simple linear regression of loss ratio on a credit-score variable, fitted to $n = 20$ territories, gives $\hat{\beta}_1 = 0.42$ with standard error $0.15$ . Test $\beta_1 = 0$ against a two-sided alternative, and give a 95% confidence interval for $\beta_1$ .
Answer $t = 0.42/0.15 = 2.8$ , with $20 - 2 = 18$ degrees of freedom. In the $df = 18$ row the two-sided $0.02$ column is $2.552$ and the $0.01$ column is $2.878$ , so the p-value is between 0.01 and 0.02: significant at 5% and 2%, not at 1%.

The two-sided $0.05$ column gives $2.101$ :
 $\begin{align*} \hat{\beta}_1 \pm 2.101\,\text{SE} &= 0.42 \pm 2.101(0.15) \\ &= 0.42 \pm 0.315 \end{align*}$ 
The interval $(0.105,\ 0.735)$ excludes 0, consistent with rejecting $\beta_1 = 0$ at 5%.