Is a goodness of fit test one-tailed or two-tailed?

One-tailed. Only a large X^2 is evidence against the model.

When must cells be combined?

When any expected frequency is less than 5. Add neighbouring cells, both Ei and Oi.

Degrees of freedom for goodness of fit?

\nu=k-1-m, with k cells after combining and m parameters estimated.

How are expected frequencies found for a continuous uniform distribution on [a,b]?

E=n\times\frac{class width}{b-a} for each class.

How is the parameter estimated for a Poisson fit?

\lambda is the sample mean. Use 'r or more' for the last cell.

What is the decision rule?

Reject H0 if X^2 exceeds the critical value of \chi^2\nu; otherwise there is insufficient evidence to reject it.

What does a very small X^2 tell you?

The observed frequencies are very close to the model, so there is no evidence against the fit.

Chi-squared goodness of fit testsEdexcel International A Level Further Maths: Flashcards

Card 1 of 130 of 13 known

Question

What are the hypotheses in a goodness of fit test?

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

What are the hypotheses in a goodness of fit test?: $H_0$ : the stated model fits the data. $H_1$ : it does not fit.
State the $\chi^2$ test statistic.: $X^2=\sum\frac{(O_i-E_i)^2}{E_i}$
Is a goodness of fit test one-tailed or two-tailed?: One-tailed. Only a large $X^2$ is evidence against the model.
When must cells be combined?: When any expected frequency is less than 5. Add neighbouring cells, both $E_i$ and $O_i$ .
Degrees of freedom for goodness of fit?: $\nu=k-1-m$ , with $k$ cells after combining and $m$ parameters estimated.
Degrees of freedom: Poisson with $\lambda$ estimated and 5 cells?: $\nu=5-1-1=3$
Degrees of freedom: Normal with $\mu$ and $\sigma^2$ both estimated and 6 cells?: $\nu=6-1-2=3$
How is $E_i$ found for a fair die thrown $n$ times?: $E_i=\frac{n}{6}$ for every face.
How are expected frequencies found for a continuous uniform distribution on $[a,b]$ ?: $E=n\times\frac{\text{class width}}{b-a}$ for each class.
How is the parameter estimated for a binomial $B(n,p)$ fit?: $\hat p=\frac{\text{sample mean}}{n}$
How is the parameter estimated for a Poisson fit?: $\lambda$ is the sample mean. Use ' $r$ or more' for the last cell.
What is the decision rule?: Reject $H_0$ if $X^2$ exceeds the critical value of $\chi^2_\nu$ ; otherwise there is insufficient evidence to reject it.
What does a very small $X^2$ tell you?: The observed frequencies are very close to the model, so there is no evidence against the fit.

Exam questions on Chi-squared goodness of fit tests

A die is thrown 120 times. Faces 1 to 6 occurred 14, 25, 17, 22, 24 and 18 times respectively. A $\chi^2$ goodness of fit test is to be used to test whether the die is fair.
Calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$ .2 marks
A firm records the number of machine breakdowns in each of 100 weeks. No breakdowns occurred in 27 weeks, 1 in 35 weeks, 2 in 22 weeks, 3 in 11 weeks, 4 in 4 weeks and 5 in 1 week. The mean number of breakdowns per week is 1.33. The firm models the weekly number of breakdowns by a Poisson distribution with mean 1.33. The expected frequencies are 26.45, 35.18, 23.39, 10.37, 3.45 and 1.17 for 0, 1, 2, 3, 4 and 5 or more breakdowns respectively.
Calculate the value of the test statistic for the combined cells.2 marks
An inspector tests 120 boxes, each containing 4 bulbs, and records the number of defective bulbs in each box. There were 48 boxes with no defective bulbs, 42 with 1, 22 with 2, 7 with 3 and 1 with 4. The inspector suggests that the number of defective bulbs in a box can be modelled by $B(4,p)$ , with $p$ estimated from the data.
State suitable hypotheses and show that the estimate of $p$ is 0.231 to 3 significant figures.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).